the chain rule (2.4) october 23rd, 2012. i. the chain rule thm. 2.10: the chain rule: if y = f(u) is...
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The chain rule (2.4)The chain rule (2.4)The chain rule (2.4)The chain rule (2.4)October 23rd, 2012October 23rd, 2012
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I. the chain ruleI. the chain rule
Thm. 2.10: The Chain Rule: If y = f(u) is a differentiable
function of u and u = g(x) is a differentiable function of x,
then y=f(g(x)) is a differentiable function of x and
-or-
Thm. 2.10: The Chain Rule: If y = f(u) is a differentiable
function of u and u = g(x) is a differentiable function of x,
then y=f(g(x)) is a differentiable function of x and
-or-
dy
dx=dydu
⋅dudx
d
dxf (g(x))[ ] = f '(g(x))g'(x)
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Ex. 1: For each function, find the derivative by using the
rules learned previously in sections 2.2 and 2.3. Then find
the derivative using the chain rule. Which method is
easier?
a.
b.
c.
Ex. 1: For each function, find the derivative by using the
rules learned previously in sections 2.2 and 2.3. Then find
the derivative using the chain rule. Which method is
easier?
a.
b.
c.
f (x)=2
3x+1
f (x)=(x+2)3
f (x)=sin2x
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II. The general power ruleII. The general power rule
Thm. 2.11: The General Power Rule: If
, where u is a differentiable function of x and n is a
rational number, then
-or-
Thm. 2.11: The General Power Rule: If
, where u is a differentiable function of x and n is a
rational number, then
-or-
y= u(x)[ ]n
dy
dx=n u(x)[ ]
n−1 dudx
d
dxun⎡⎣ ⎤⎦=nu
n−1u'
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A. Applying the general power ruleA. Applying the general power rule
Ex. 2: Find the derivative of .Ex. 2: Find the derivative of .f (x)=(4x−3x2 )4
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You Try: Find the derivative of .You Try: Find the derivative of .f (x)=(5x2 +2x5 )7
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B. Differentiating functions involving radicalsB. Differentiating functions involving radicals
Ex. 3: Find all the points on the graph of
for which f ’(x) = 0 and those for which f ‘(x) does not
exist.
Ex. 3: Find all the points on the graph of
for which f ’(x) = 0 and those for which f ‘(x) does not
exist.
f (x)= (3x2 −5)23
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You Try: Find all the points on the graph of
for which f ‘(x) = 0 and those for which f ‘(x) does not
exist.
You Try: Find all the points on the graph of
for which f ‘(x) = 0 and those for which f ‘(x) does not
exist.
f (x)= (4x−3)3
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C. differentiating quotients with constant numeratorsC. differentiating quotients with constant numerators
Ex. 4: Differentiate .Ex. 4: Differentiate .f (x)=−5
(5x−4)3
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You Try: Differentiate .You Try: Differentiate .g(t)=−3
(4x+6)2
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III. Simplifying derivativesIII. Simplifying derivatives
Ex. 5: Find the derivative of .Ex. 5: Find the derivative of .f (x)=x3 2−3x2
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You Try: Find the derivative of .You Try: Find the derivative of .f (x)=x2 5−2x2
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Ex. 6: Find the derivative of .Ex. 6: Find the derivative of .f (x)=2x
5x2 −33
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You Try: Find the derivative of .You Try: Find the derivative of .f (x)=x
x2 −63
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Ex. 7: Find the derivative of .Ex. 7: Find the derivative of .f (x)=4x−9x2 +2
⎛⎝⎜
⎞⎠⎟
2
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You Try: Find the derivative of .You Try: Find the derivative of .f (x)=2x+63x2 +1
⎛⎝⎜
⎞⎠⎟
2
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IV. trigonometric functions & the chain ruleIV. trigonometric functions & the chain rule
A. Applying the Chain Rule to Trigonometric Functions
Ex. 8: Find the derivative of each function.
a. y = cos 2x
b. y = sin (4x+1)
c. y = cot 4x
A. Applying the Chain Rule to Trigonometric Functions
Ex. 8: Find the derivative of each function.
a. y = cos 2x
b. y = sin (4x+1)
c. y = cot 4x
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You Try: Find the derivative of each function.
a.
b. y = csc 4x + 1
c.
d.
You Try: Find the derivative of each function.
a.
b. y = csc 4x + 1
c.
d.
y=sin4x2
y=sin(4x)2
y= sinx
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B. repeated application of the chain ruleB. repeated application of the chain rule
Ex. 9: Find the derivative of .Ex. 9: Find the derivative of .f (t)=cos3 5t
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You Try: Find the derivative of .You Try: Find the derivative of .f (t)=sin2(6t−3)
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C. tangent line of a trigonometric functionC. tangent line of a trigonometric function
Ex. 10: Find an equation of the tangent line to the graph
of
at the point .
Ex. 10: Find an equation of the tangent line to the graph
of
at the point .y=2 tan3 x π
4,2( )