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More on Derivatives and Integrals
-Product Rule-Chain Rule
AP Physics CMrs. Coyle
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f’ (x) = lim f(x + h)f(x ) h 0h
Derivative
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Derivative Notations
f’ (x)
df (x) dx
. f
dfdx
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Notations when evaluating the derivative at x=a
f(a)
df (a) dx
f’(a)
df |x=a
dx
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Basic Derivatives d(c) = 0 dx
d(mx+b) = m dx
d(x n) = n x n-1
dx n is any integer
x≠0
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Derivative of a polynomial.
For y(x) = axn
dy = a n xn-1
dx
-Apply to each term of the polynomial.
-Note that the derivative of constant is 0.
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Product Rule
For two functions of x: u(x) and v (x)
d [u(x) v (x)] =u d v (x) + v d u (x) dx dx dx
or
(uv)’ = u v’ + vu’
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Example of Product Rule:
Differentiate: F=(3x-2)(x2 + 5x + 1)
Answer: F’(x) = 9x2 + 26x-7
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Chain Rule
If y=f(u) and u=g(x):
dy = dy dudx du dx
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Example of Chain RuleDifferentiate: F(x)= (x 2 + 1) 3
Ans:F’(x)= 6(x2 +1)2x
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Second Derivative Notations df’ (x) dx
d2f (x) d x2
f’’(x)
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Example of Second Derivative
Compute the second derivative of y=(x)1/2
Ans: (-1/4) x-3/2
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Derivatives of Trig Functions
d sinx = cosx dx
d cosx = -sinx dx
d tanx = sec2 x dx
d secx = secx tanx dx
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Derivative of the Exponential Function
d e u = e u du dx dx
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Example of derivative of Exponential Function
2
Differentiate: e x
2
Ans: 2x e x
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Derivative of Ln
d (lnx) = 1/x dx
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Definite Integral
b
a∫b f(x) dx= F(b)-F(a)= F(x)|a
a and b are the limits of integration.
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If F(x)= ∫ f(x) dx
then
d F(x) = f(x) dx
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Properties of Integrals
a∫c f(x) dx = a∫b f(x) dx+ b∫c f(x) dx
a<b<c
a∫b (f(x)+g(x)) dx = a∫b f(x) dx+ a∫b g(x) dx
a∫b cf(x) dx =c a∫b f(x) dx
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Basic Integrals (integration constant ommited)
∫ xn dx = 1 xn+1 , n ≠ 1 n+1
∫ ex dx = ex
∫ (1/x) dx = ln|x|∫ cosx dx = sinx
∫ (1/x) dx = ln|x|∫ sinx dx = -cosx
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Example with computing work.
• There is a force of 5x2 –x +2 N pulling on an object. Compute the work done in moving it from x=1m to x=4m.
• Ans: 103.5N
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To evaluate integrals of products of functions :
• Chain Rule
• Integration by parts
• Change of Variable Formula
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Change of Variable Formula
When a function and its derivative appear in the integral:
a∫b f[g(x)]g’(x) dx = g(a)∫g(b) f(y) dy
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Example: When a function and its derivative appear in the integral:
• Compute x=0∫x=1 2x (x2 +1) 3 dx
• Ans: 3.75
• Ans:
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Example of Change of Variable Formula
Evaluate: 0∫1 2x (x2 + 1) 9 dx
Answ: 102.3
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Integration by Parts
a∫b u(x) dv dx=
dx
b
= u(x) v(x)|a - a∫b v(x) du dx
dx
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Integration by Parts
b
a∫b u v’ dx= u v|a - a∫b v u’ dx
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Example of Integration by Parts
Compute 0∫π x sinx dx
Ans: π