lesson 8: basic differentiation rules
DESCRIPTION
TRANSCRIPT
Section 2.3Basic Differentiation Rules
V63.0121.002.2010Su, Calculus I
New York University
May 24, 2010
Announcements
I Homework 1 due TuesdayI Quiz 2 Thursday in class on Sections 1.5–2.5
. . . . . .
. . . . . .
Announcements
I Homework 1 due TuesdayI Quiz 2 Thursday in class
on Sections 1.5–2.5
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 2 / 36
. . . . . .
Objectives
I Understand and use thesedifferentiation rules:
I the derivative of aconstant function (zero);
I the Constant MultipleRule;
I the Sum Rule;I the Difference Rule;I the derivatives of sine
and cosine.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 3 / 36
. . . . . .
Outline
Derivatives so farDerivatives of power functions by handThe Power Rule
Derivatives of polynomialsThe Power Rule for whole number powersThe Power Rule for constantsThe Sum RuleThe Constant Multiple Rule
Derivatives of sine and cosine
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 4 / 36
. . . . . .
Derivative of the squaring function
Example
Suppose f(x) = x2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2 − x2
h
= limh→0
��x2 + 2xh+ h2 −��x2
h= lim
h→0
2x�h+ h�2
�h= lim
h→0(2x+ h) = 2x.
So f′(x) = 2x.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 5 / 36
. . . . . .
Derivative of the squaring function
Example
Suppose f(x) = x2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2 − x2
h
= limh→0
��x2 + 2xh+ h2 −��x2
h= lim
h→0
2x�h+ h�2
�h= lim
h→0(2x+ h) = 2x.
So f′(x) = 2x.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 5 / 36
. . . . . .
Derivative of the squaring function
Example
Suppose f(x) = x2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2 − x2
h
= limh→0
��x2 + 2xh+ h2 −��x2
h= lim
h→0
2x�h+ h�2
�h= lim
h→0(2x+ h) = 2x.
So f′(x) = 2x.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 5 / 36
. . . . . .
Derivative of the squaring function
Example
Suppose f(x) = x2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2 − x2
h
= limh→0
��x2 + 2xh+ h2 −��x2
h
= limh→0
2x�h+ h�2
�h= lim
h→0(2x+ h) = 2x.
So f′(x) = 2x.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 5 / 36
. . . . . .
Derivative of the squaring function
Example
Suppose f(x) = x2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2 − x2
h
= limh→0
��x2 + 2xh+ h2 −��x2
h= lim
h→0
2x�h+ h�2
�h
= limh→0
(2x+ h) = 2x.
So f′(x) = 2x.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 5 / 36
. . . . . .
Derivative of the squaring function
Example
Suppose f(x) = x2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2 − x2
h
= limh→0
��x2 + 2xh+ h2 −��x2
h= lim
h→0
2x�h+ h�2
�h= lim
h→0(2x+ h) = 2x.
So f′(x) = 2x.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 5 / 36
. . . . . .
Derivative of the cubing function
Example
Suppose f(x) = x3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)3 − x3
h
= limh→0
��x3 + 3x2h+ 3xh2 + h3 −��x3
h= lim
h→0
3x2�h+ 3xh���1
2 + h���2
3
�h= lim
h→0
(3x2 + 3xh+ h2
)= 3x2.
So f′(x) = 3x2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 6 / 36
. . . . . .
Derivative of the cubing function
Example
Suppose f(x) = x3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)3 − x3
h
= limh→0
��x3 + 3x2h+ 3xh2 + h3 −��x3
h= lim
h→0
3x2�h+ 3xh���1
2 + h���2
3
�h= lim
h→0
(3x2 + 3xh+ h2
)= 3x2.
So f′(x) = 3x2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 6 / 36
. . . . . .
Derivative of the cubing function
Example
Suppose f(x) = x3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)3 − x3
h
= limh→0
��x3 + 3x2h+ 3xh2 + h3 −��x3
h
= limh→0
3x2�h+ 3xh���1
2 + h���2
3
�h= lim
h→0
(3x2 + 3xh+ h2
)= 3x2.
So f′(x) = 3x2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 6 / 36
. . . . . .
Derivative of the cubing function
Example
Suppose f(x) = x3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)3 − x3
h
= limh→0
��x3 + 3x2h+ 3xh2 + h3 −��x3
h= lim
h→0
3x2�h+ 3xh���1
2 + h���2
3
�h
= limh→0
(3x2 + 3xh+ h2
)= 3x2.
So f′(x) = 3x2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 6 / 36
. . . . . .
Derivative of the cubing function
Example
Suppose f(x) = x3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)3 − x3
h
= limh→0
��x3 + 3x2h+ 3xh2 + h3 −��x3
h= lim
h→0
3x2�h+ 3xh���1
2 + h���2
3
�h= lim
h→0
(3x2 + 3xh+ h2
)= 3x2.
So f′(x) = 3x2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 6 / 36
. . . . . .
The cubing function and its derivatives
. .x
.y
.f
.f′.f′′I Notice that f is increasing,
and f′ > 0 except f′(0) = 0I Notice also that the
tangent line to the graph off at (0,0) crosses thegraph (contrary to apopular “definition” of thetangent line)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 7 / 36
. . . . . .
The cubing function and its derivatives
. .x
.y
.f
.f′.f′′I Notice that f is increasing,
and f′ > 0 except f′(0) = 0I Notice also that the
tangent line to the graph off at (0,0) crosses thegraph (contrary to apopular “definition” of thetangent line)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 7 / 36
. . . . . .
The cubing function and its derivatives
. .x
.y
.f
.f′
.f′′
I Notice that f is increasing,and f′ > 0 except f′(0) = 0
I Notice also that thetangent line to the graph off at (0,0) crosses thegraph (contrary to apopular “definition” of thetangent line)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 7 / 36
. . . . . .
The cubing function and its derivatives
. .x
.y
.f
.f′.f′′I Notice that f is increasing,
and f′ > 0 except f′(0) = 0
I Notice also that thetangent line to the graph off at (0,0) crosses thegraph (contrary to apopular “definition” of thetangent line)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 7 / 36
. . . . . .
The cubing function and its derivatives
. .x
.y
.f
.f′.f′′I Notice that f is increasing,
and f′ > 0 except f′(0) = 0I Notice also that the
tangent line to the graph off at (0,0) crosses thegraph (contrary to apopular “definition” of thetangent line)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 7 / 36
. . . . . .
Derivative of the square root function
Example
Suppose f(x) =√x = x1/2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
√x+ h−
√x
h
= limh→0
√x+ h−
√x
h·√x+ h+
√x√
x+ h+√x
= limh→0
(�x+ h)− �xh(√
x+ h+√x) = lim
h→0
�h�h(√
x+ h+√x)
=1
2√x
So f′(x) =√x = 1
2x−1/2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 8 / 36
. . . . . .
Derivative of the square root function
Example
Suppose f(x) =√x = x1/2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
√x+ h−
√x
h
= limh→0
√x+ h−
√x
h·√x+ h+
√x√
x+ h+√x
= limh→0
(�x+ h)− �xh(√
x+ h+√x) = lim
h→0
�h�h(√
x+ h+√x)
=1
2√x
So f′(x) =√x = 1
2x−1/2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 8 / 36
. . . . . .
Derivative of the square root function
Example
Suppose f(x) =√x = x1/2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
√x+ h−
√x
h
= limh→0
√x+ h−
√x
h·√x+ h+
√x√
x+ h+√x
= limh→0
(�x+ h)− �xh(√
x+ h+√x) = lim
h→0
�h�h(√
x+ h+√x)
=1
2√x
So f′(x) =√x = 1
2x−1/2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 8 / 36
. . . . . .
Derivative of the square root function
Example
Suppose f(x) =√x = x1/2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
√x+ h−
√x
h
= limh→0
√x+ h−
√x
h·√x+ h+
√x√
x+ h+√x
= limh→0
(�x+ h)− �xh(√
x+ h+√x)
= limh→0
�h�h(√
x+ h+√x)
=1
2√x
So f′(x) =√x = 1
2x−1/2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 8 / 36
. . . . . .
Derivative of the square root function
Example
Suppose f(x) =√x = x1/2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
√x+ h−
√x
h
= limh→0
√x+ h−
√x
h·√x+ h+
√x√
x+ h+√x
= limh→0
(�x+ h)− �xh(√
x+ h+√x) = lim
h→0
�h�h(√
x+ h+√x)
=1
2√x
So f′(x) =√x = 1
2x−1/2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 8 / 36
. . . . . .
Derivative of the square root function
Example
Suppose f(x) =√x = x1/2. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
√x+ h−
√x
h
= limh→0
√x+ h−
√x
h·√x+ h+
√x√
x+ h+√x
= limh→0
(�x+ h)− �xh(√
x+ h+√x) = lim
h→0
�h�h(√
x+ h+√x)
=1
2√x
So f′(x) =√x = 1
2x−1/2.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 8 / 36
. . . . . .
The square root function and its derivatives
. .x
.y
.f.f′
I Here limx→0+
f′(x) = ∞ and f
is not differentiable at 0I Notice also lim
x→∞f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 9 / 36
. . . . . .
The square root function and its derivatives
. .x
.y
.f
.f′I Here lim
x→0+f′(x) = ∞ and f
is not differentiable at 0I Notice also lim
x→∞f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 9 / 36
. . . . . .
The square root function and its derivatives
. .x
.y
.f.f′
I Here limx→0+
f′(x) = ∞ and f
is not differentiable at 0
I Notice also limx→∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 9 / 36
. . . . . .
The square root function and its derivatives
. .x
.y
.f.f′
I Here limx→0+
f′(x) = ∞ and f
is not differentiable at 0I Notice also lim
x→∞f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 9 / 36
. . . . . .
Derivative of the cube root function
Example
Suppose f(x) = 3√x = x1/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)1/3 − x1/3
h
= limh→0
(x+ h)1/3 − x1/3
h· (x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
(x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
= limh→0
(�x+ h)− �xh((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)= lim
h→0
�h�h((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
) =1
3x2/3
So f′(x) = 13x
−2/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 10 / 36
. . . . . .
Derivative of the cube root function
Example
Suppose f(x) = 3√x = x1/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)1/3 − x1/3
h
= limh→0
(x+ h)1/3 − x1/3
h· (x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
(x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
= limh→0
(�x+ h)− �xh((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)= lim
h→0
�h�h((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
) =1
3x2/3
So f′(x) = 13x
−2/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 10 / 36
. . . . . .
Derivative of the cube root function
Example
Suppose f(x) = 3√x = x1/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)1/3 − x1/3
h
= limh→0
(x+ h)1/3 − x1/3
h· (x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
(x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
= limh→0
(�x+ h)− �xh((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)= lim
h→0
�h�h((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
) =1
3x2/3
So f′(x) = 13x
−2/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 10 / 36
. . . . . .
Derivative of the cube root function
Example
Suppose f(x) = 3√x = x1/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)1/3 − x1/3
h
= limh→0
(x+ h)1/3 − x1/3
h· (x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
(x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
= limh→0
(�x+ h)− �xh((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)
= limh→0
�h�h((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
) =1
3x2/3
So f′(x) = 13x
−2/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 10 / 36
. . . . . .
Derivative of the cube root function
Example
Suppose f(x) = 3√x = x1/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)1/3 − x1/3
h
= limh→0
(x+ h)1/3 − x1/3
h· (x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
(x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
= limh→0
(�x+ h)− �xh((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)= lim
h→0
�h�h((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)
=1
3x2/3
So f′(x) = 13x
−2/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 10 / 36
. . . . . .
Derivative of the cube root function
Example
Suppose f(x) = 3√x = x1/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)1/3 − x1/3
h
= limh→0
(x+ h)1/3 − x1/3
h· (x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
(x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
= limh→0
(�x+ h)− �xh((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
)= lim
h→0
�h�h((x+ h)2/3 + (x+ h)1/3x1/3 + x2/3
) =1
3x2/3
So f′(x) = 13x
−2/3.V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 10 / 36
. . . . . .
The cube root function and its derivatives
. .x
.y
.f.f′
I Here limx→0
f′(x) = ∞ and f isnot differentiable at 0
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 11 / 36
. . . . . .
The cube root function and its derivatives
. .x
.y
.f
.f′
I Here limx→0
f′(x) = ∞ and f isnot differentiable at 0
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 11 / 36
. . . . . .
The cube root function and its derivatives
. .x
.y
.f.f′
I Here limx→0
f′(x) = ∞ and f isnot differentiable at 0
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 11 / 36
. . . . . .
The cube root function and its derivatives
. .x
.y
.f.f′
I Here limx→0
f′(x) = ∞ and f isnot differentiable at 0
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 11 / 36
. . . . . .
One more
Example
Suppose f(x) = x2/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2/3 − x2/3
h
= limh→0
(x+ h)1/3 − x1/3
h·((x+ h)1/3 + x1/3
)= 1
3x−2/3
(2x1/3
)= 2
3x−1/3
So f′(x) = 23x
−1/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 12 / 36
. . . . . .
One more
Example
Suppose f(x) = x2/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2/3 − x2/3
h
= limh→0
(x+ h)1/3 − x1/3
h·((x+ h)1/3 + x1/3
)= 1
3x−2/3
(2x1/3
)= 2
3x−1/3
So f′(x) = 23x
−1/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 12 / 36
. . . . . .
One more
Example
Suppose f(x) = x2/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2/3 − x2/3
h
= limh→0
(x+ h)1/3 − x1/3
h·((x+ h)1/3 + x1/3
)
= 13x
−2/3(2x1/3
)= 2
3x−1/3
So f′(x) = 23x
−1/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 12 / 36
. . . . . .
One more
Example
Suppose f(x) = x2/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2/3 − x2/3
h
= limh→0
(x+ h)1/3 − x1/3
h·((x+ h)1/3 + x1/3
)= 1
3x−2/3
(2x1/3
)
= 23x
−1/3
So f′(x) = 23x
−1/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 12 / 36
. . . . . .
One more
Example
Suppose f(x) = x2/3. Use the definition of derivative to find f′(x).
Solution
f′(x) = limh→0
f(x+ h)− f(x)h
= limh→0
(x+ h)2/3 − x2/3
h
= limh→0
(x+ h)1/3 − x1/3
h·((x+ h)1/3 + x1/3
)= 1
3x−2/3
(2x1/3
)= 2
3x−1/3
So f′(x) = 23x
−1/3.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 12 / 36
. . . . . .
The function x 7→ x2/3 and its derivative
. .x
.y
.f.f′
I f is not differentiable at 0and lim
x→0±f′(x) = ±∞
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 13 / 36
. . . . . .
The function x 7→ x2/3 and its derivative
. .x
.y
.f
.f′
I f is not differentiable at 0and lim
x→0±f′(x) = ±∞
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 13 / 36
. . . . . .
The function x 7→ x2/3 and its derivative
. .x
.y
.f.f′
I f is not differentiable at 0and lim
x→0±f′(x) = ±∞
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 13 / 36
. . . . . .
The function x 7→ x2/3 and its derivative
. .x
.y
.f.f′
I f is not differentiable at 0and lim
x→0±f′(x) = ±∞
I Notice also limx→±∞
f′(x) = 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 13 / 36
. . . . . .
Recap
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap
y y′
x2 2x1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
Recap: The Tower of Power
y y′
x2 2x
1
x3 3x2
x1/2 12x
−1/2
x1/3 13x
−2/3
x2/3 23x
−1/3
I The power goes down byone in each derivative
I The coefficient in thederivative is the power ofthe original function
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 14 / 36
. . . . . .
The Power Rule
There is mounting evidence for
Theorem (The Power Rule)
Let r be a real number and f(x) = xr. Then
f′(x) = rxr−1
as long as the expression on the right-hand side is defined.
I Perhaps the most famous rule in calculusI We will assume it as of todayI We will prove it many ways for many different r.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 15 / 36
. . . . . .
The other Tower of Power
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 16 / 36
. . . . . .
Outline
Derivatives so farDerivatives of power functions by handThe Power Rule
Derivatives of polynomialsThe Power Rule for whole number powersThe Power Rule for constantsThe Sum RuleThe Constant Multiple Rule
Derivatives of sine and cosine
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 17 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that. The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that. The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that.
The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that. The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that. The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that. The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Remember your algebra
FactLet n be a positive whole number. Then
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
Proof.We have
(x+ h)n = (x+ h) · (x+ h) · (x+ h) · · · (x+ h)︸ ︷︷ ︸n copies
=n∑
k=0
ckxkhn−k
The coefficient of xn is 1 because we have to choose x from eachbinomial, and there’s only one way to do that. The coefficient of xn−1his the number of ways we can choose x n− 1 times, which is the sameas the number of different hs we can pick, which is n.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 18 / 36
. . . . . .
Pascal's Triangle
..1
.1 .1
.1 .2 .1
.1 .3 .3 .1
.1 .4 .6 .4 .1
.1 .5 .10 .10 .5 .1
.1 .6 .15 .20 .15 .6 .1
(x+ h)0 = 1
(x+ h)1 = 1x+ 1h
(x+ h)2 = 1x2 + 2xh+ 1h2
(x+ h)3 = 1x3 + 3x2h+ 3xh2 + 1h3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 19 / 36
. . . . . .
Pascal's Triangle
..1
.1 .1
.1 .2 .1
.1 .3 .3 .1
.1 .4 .6 .4 .1
.1 .5 .10 .10 .5 .1
.1 .6 .15 .20 .15 .6 .1
(x+ h)0 = 1
(x+ h)1 = 1x+ 1h
(x+ h)2 = 1x2 + 2xh+ 1h2
(x+ h)3 = 1x3 + 3x2h+ 3xh2 + 1h3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 19 / 36
. . . . . .
Pascal's Triangle
..1
.1 .1
.1 .2 .1
.1 .3 .3 .1
.1 .4 .6 .4 .1
.1 .5 .10 .10 .5 .1
.1 .6 .15 .20 .15 .6 .1
(x+ h)0 = 1
(x+ h)1 = 1x+ 1h
(x+ h)2 = 1x2 + 2xh+ 1h2
(x+ h)3 = 1x3 + 3x2h+ 3xh2 + 1h3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 19 / 36
. . . . . .
Pascal's Triangle
..1
.1 .1
.1 .2 .1
.1 .3 .3 .1
.1 .4 .6 .4 .1
.1 .5 .10 .10 .5 .1
.1 .6 .15 .20 .15 .6 .1
(x+ h)0 = 1
(x+ h)1 = 1x+ 1h
(x+ h)2 = 1x2 + 2xh+ 1h2
(x+ h)3 = 1x3 + 3x2h+ 3xh2 + 1h3
. . . . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 19 / 36
. . . . . .
Theorem (The Power Rule)
Let r be a positive whole number. Then
ddx
xr = rxr−1
Proof.As we showed above,
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
So
(x+ h)n − xn
h=
nxn−1h+ (stuff with at least two hs in it)h
= nxn−1 + (stuff with at least one h in it)
and this tends to nxn−1 as h → 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 20 / 36
. . . . . .
Theorem (The Power Rule)
Let r be a positive whole number. Then
ddx
xr = rxr−1
Proof.As we showed above,
(x+ h)n = xn + nxn−1h+ (stuff with at least two hs in it)
So
(x+ h)n − xn
h=
nxn−1h+ (stuff with at least two hs in it)h
= nxn−1 + (stuff with at least one h in it)
and this tends to nxn−1 as h → 0.V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 20 / 36
. . . . . .
The Power Rule for constants
TheoremLet c be a constant. Then
ddx
c = 0
.
.likeddx
x0 = 0x−1
(although x 7→ 0x−1 is not defined at zero.)
Proof.Let f(x) = c. Then
f(x+ h)− f(x)h
=c− ch
= 0
So f′(x) = limh→0
0 = 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 21 / 36
. . . . . .
The Power Rule for constants
TheoremLet c be a constant. Then
ddx
c = 0 .
.likeddx
x0 = 0x−1
(although x 7→ 0x−1 is not defined at zero.)
Proof.Let f(x) = c. Then
f(x+ h)− f(x)h
=c− ch
= 0
So f′(x) = limh→0
0 = 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 21 / 36
. . . . . .
The Power Rule for constants
TheoremLet c be a constant. Then
ddx
c = 0 .
.likeddx
x0 = 0x−1
(although x 7→ 0x−1 is not defined at zero.)
Proof.Let f(x) = c. Then
f(x+ h)− f(x)h
=c− ch
= 0
So f′(x) = limh→0
0 = 0.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 21 / 36
. . . . . .
Recall the Limit Laws
FactSuppose lim
x→af(x) = L and lim
x→ag(x) = M and c is a constant. Then
1. limx→a
[f(x) + g(x)] = L+M
2. limx→a
[f(x)− g(x)] = L−M
3. limx→a
[cf(x)] = cL
4. . . .
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 22 / 36
. . . . . .
Adding functions
Theorem (The Sum Rule)
Let f and g be functions and define
(f+ g)(x) = f(x) + g(x)
Then if f and g are differentiable at x, then so is f+ g and
(f+ g)′(x) = f′(x) + g′(x).
Succinctly, (f+ g)′ = f′ + g′.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 23 / 36
. . . . . .
Proof.Follow your nose:
(f+ g)′(x) = limh→0
(f+ g)(x+ h)− (f+ g)(x)h
= limh→0
f(x+ h) + g(x+ h)− [f(x) + g(x)]h
= limh→0
f(x+ h)− f(x)h
+ limh→0
g(x+ h)− g(x)h
= f′(x) + g′(x)
Note the use of the Sum Rule for limits. Since the limits of thedifference quotients for for f and g exist, the limit of the sum is the sumof the limits.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 24 / 36
. . . . . .
Proof.Follow your nose:
(f+ g)′(x) = limh→0
(f+ g)(x+ h)− (f+ g)(x)h
= limh→0
f(x+ h) + g(x+ h)− [f(x) + g(x)]h
= limh→0
f(x+ h)− f(x)h
+ limh→0
g(x+ h)− g(x)h
= f′(x) + g′(x)
Note the use of the Sum Rule for limits. Since the limits of thedifference quotients for for f and g exist, the limit of the sum is the sumof the limits.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 24 / 36
. . . . . .
Proof.Follow your nose:
(f+ g)′(x) = limh→0
(f+ g)(x+ h)− (f+ g)(x)h
= limh→0
f(x+ h) + g(x+ h)− [f(x) + g(x)]h
= limh→0
f(x+ h)− f(x)h
+ limh→0
g(x+ h)− g(x)h
= f′(x) + g′(x)
Note the use of the Sum Rule for limits. Since the limits of thedifference quotients for for f and g exist, the limit of the sum is the sumof the limits.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 24 / 36
. . . . . .
Proof.Follow your nose:
(f+ g)′(x) = limh→0
(f+ g)(x+ h)− (f+ g)(x)h
= limh→0
f(x+ h) + g(x+ h)− [f(x) + g(x)]h
= limh→0
f(x+ h)− f(x)h
+ limh→0
g(x+ h)− g(x)h
= f′(x) + g′(x)
Note the use of the Sum Rule for limits. Since the limits of thedifference quotients for for f and g exist, the limit of the sum is the sumof the limits.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 24 / 36
. . . . . .
Scaling functions
Theorem (The Constant Multiple Rule)
Let f be a function and c a constant. Define
(cf)(x) = cf(x)
Then if f is differentiable at x, so is cf and
(cf)′(x) = c · f′(x)
Succinctly, (cf)′ = cf′.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 25 / 36
. . . . . .
Proof.Again, follow your nose.
(cf)′(x) = limh→0
(cf)(x+ h)− (cf)(x)h
= limh→0
cf(x+ h)− cf(x)h
= c limh→0
f(x+ h)− f(x)h
= c · f′(x)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 26 / 36
. . . . . .
Proof.Again, follow your nose.
(cf)′(x) = limh→0
(cf)(x+ h)− (cf)(x)h
= limh→0
cf(x+ h)− cf(x)h
= c limh→0
f(x+ h)− f(x)h
= c · f′(x)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 26 / 36
. . . . . .
Proof.Again, follow your nose.
(cf)′(x) = limh→0
(cf)(x+ h)− (cf)(x)h
= limh→0
cf(x+ h)− cf(x)h
= c limh→0
f(x+ h)− f(x)h
= c · f′(x)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 26 / 36
. . . . . .
Proof.Again, follow your nose.
(cf)′(x) = limh→0
(cf)(x+ h)− (cf)(x)h
= limh→0
cf(x+ h)− cf(x)h
= c limh→0
f(x+ h)− f(x)h
= c · f′(x)
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 26 / 36
. . . . . .
Derivatives of polynomials
Example
Findddx
(2x3 + x4 − 17x12 + 37
)
Solution
ddx
(2x3 + x4 − 17x12 + 37
)=
ddx
(2x3
)+
ddx
x4 +ddx
(−17x12
)+
ddx
(37)
= 2ddx
x3 +ddx
x4 − 17ddx
x12 + 0
= 2 · 3x2 + 4x3 − 17 · 12x11
= 6x2 + 4x3 − 204x11
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 27 / 36
. . . . . .
Derivatives of polynomials
Example
Findddx
(2x3 + x4 − 17x12 + 37
)Solution
ddx
(2x3 + x4 − 17x12 + 37
)=
ddx
(2x3
)+
ddx
x4 +ddx
(−17x12
)+
ddx
(37)
= 2ddx
x3 +ddx
x4 − 17ddx
x12 + 0
= 2 · 3x2 + 4x3 − 17 · 12x11
= 6x2 + 4x3 − 204x11
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 27 / 36
. . . . . .
Derivatives of polynomials
Example
Findddx
(2x3 + x4 − 17x12 + 37
)Solution
ddx
(2x3 + x4 − 17x12 + 37
)=
ddx
(2x3
)+
ddx
x4 +ddx
(−17x12
)+
ddx
(37)
= 2ddx
x3 +ddx
x4 − 17ddx
x12 + 0
= 2 · 3x2 + 4x3 − 17 · 12x11
= 6x2 + 4x3 − 204x11
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 27 / 36
. . . . . .
Derivatives of polynomials
Example
Findddx
(2x3 + x4 − 17x12 + 37
)Solution
ddx
(2x3 + x4 − 17x12 + 37
)=
ddx
(2x3
)+
ddx
x4 +ddx
(−17x12
)+
ddx
(37)
= 2ddx
x3 +ddx
x4 − 17ddx
x12 + 0
= 2 · 3x2 + 4x3 − 17 · 12x11
= 6x2 + 4x3 − 204x11
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 27 / 36
. . . . . .
Derivatives of polynomials
Example
Findddx
(2x3 + x4 − 17x12 + 37
)Solution
ddx
(2x3 + x4 − 17x12 + 37
)=
ddx
(2x3
)+
ddx
x4 +ddx
(−17x12
)+
ddx
(37)
= 2ddx
x3 +ddx
x4 − 17ddx
x12 + 0
= 2 · 3x2 + 4x3 − 17 · 12x11
= 6x2 + 4x3 − 204x11
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 27 / 36
. . . . . .
Outline
Derivatives so farDerivatives of power functions by handThe Power Rule
Derivatives of polynomialsThe Power Rule for whole number powersThe Power Rule for constantsThe Sum RuleThe Constant Multiple Rule
Derivatives of sine and cosine
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 28 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = ???
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1 = cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 29 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = ???
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1 = cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 29 / 36
. . . . . .
Angle addition formulasSee Appendix A
.
.
sin(A+ B) = sinA cosB+ cosA sinBcos(A+ B) = cosA cosB− sinA sinB
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 30 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = ???
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1 = cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 31 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = ???
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1 = cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 31 / 36
. . . . . .
Two important trigonometric limitsSee Section 1.4
..θ
.sin θ
.1− cos θ
.θ
.−1 .1
.
.
limθ→0
sin θθ
= 1
limθ→0
cos θ − 1θ
= 0
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 32 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = ???
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1
= cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 33 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = ???
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1
= cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 33 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = cos x
Proof.From the definition:
ddx
sin x = limh→0
sin(x+ h)− sin xh
= limh→0
( sin x cos h+ cos x sin h)− sin xh
= sin x · limh→0
cos h− 1h
+ cos x · limh→0
sin hh
= sin x · 0+ cos x · 1 = cos x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 33 / 36
. . . . . .
Illustration of Sine and Cosine
. .x
.y
.π .−π2 .0 .π2 .π
.sin x
.cos x
I f(x) = sin x has horizontal tangents where f′ = cos(x) is zero.I what happens at the horizontal tangents of cos?
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 34 / 36
. . . . . .
Illustration of Sine and Cosine
. .x
.y
.π .−π2 .0 .π2 .π
.sin x
.cos x
I f(x) = sin x has horizontal tangents where f′ = cos(x) is zero.
I what happens at the horizontal tangents of cos?
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 34 / 36
. . . . . .
Illustration of Sine and Cosine
. .x
.y
.π .−π2 .0 .π2 .π
.sin x
.cos x
I f(x) = sin x has horizontal tangents where f′ = cos(x) is zero.I what happens at the horizontal tangents of cos?
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 34 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = cos xddx
cos x = − sin x
Proof.We already did the first. The second is similar (mutatis mutandis):
ddx
cos x = limh→0
cos(x+ h)− cos xh
= limh→0
(cos x cosh− sin x sinh)− cos xh
= cos x · limh→0
cosh− 1h
− sin x · limh→0
sin hh
= cos x · 0− sin x · 1 = − sin x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 35 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = cos xddx
cos x = − sin x
Proof.We already did the first. The second is similar (mutatis mutandis):
ddx
cos x = limh→0
cos(x+ h)− cos xh
= limh→0
(cos x cosh− sin x sinh)− cos xh
= cos x · limh→0
cosh− 1h
− sin x · limh→0
sin hh
= cos x · 0− sin x · 1 = − sin x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 35 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = cos xddx
cos x = − sin x
Proof.We already did the first. The second is similar (mutatis mutandis):
ddx
cos x = limh→0
cos(x+ h)− cos xh
= limh→0
(cos x cosh− sin x sinh)− cos xh
= cos x · limh→0
cosh− 1h
− sin x · limh→0
sin hh
= cos x · 0− sin x · 1 = − sin x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 35 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = cos xddx
cos x = − sin x
Proof.We already did the first. The second is similar (mutatis mutandis):
ddx
cos x = limh→0
cos(x+ h)− cos xh
= limh→0
(cos x cosh− sin x sinh)− cos xh
= cos x · limh→0
cosh− 1h
− sin x · limh→0
sin hh
= cos x · 0− sin x · 1 = − sin x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 35 / 36
. . . . . .
Derivatives of Sine and Cosine
Fact
ddx
sin x = cos xddx
cos x = − sin x
Proof.We already did the first. The second is similar (mutatis mutandis):
ddx
cos x = limh→0
cos(x+ h)− cos xh
= limh→0
(cos x cosh− sin x sinh)− cos xh
= cos x · limh→0
cosh− 1h
− sin x · limh→0
sin hh
= cos x · 0− sin x · 1 = − sin x
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 35 / 36
. . . . . .
Summary
I Power Rule:ddx
xr = rxr−1
I Sum, Difference, Constant, and Constant Multiple RulesI Derivative of sin is cos. Derivative of cos is − sin.
V63.0121.002.2010Su, Calculus I (NYU) Section 2.3 Basic Differentiation Rules May 24, 2010 36 / 36