More on Derivatives and Integrals

-Product Rule-Chain Rule

AP Physics CMrs. Coyle

f’ (x) = lim f(x + h)f(x ) h 0h

Derivative

Derivative Notations

f’ (x)

df (x) dx

. f

dfdx

Notations when evaluating the derivative at x=a

f(a)

df (a) dx

f’(a)

df |x=a

dx

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

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.

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’

Example of Product Rule:

Differentiate: F=(3x-2)(x2 + 5x + 1)

Answer: F’(x) = 9x2 + 26x-7

Chain Rule

If y=f(u) and u=g(x):

dy = dy dudx du dx

Example of Chain RuleDifferentiate: F(x)= (x 2 + 1) 3

Ans:F’(x)= 6(x2 +1)2x

Second Derivative Notations

df’ (x) dx

d2f (x) d x2

f’’(x)

Example of Second Derivative

Compute the second derivative of y=(x)1/2

Ans: (-1/4) x-3/2

Derivatives of Trig Functions

d sinx = cosx dx

d cosx = -sinx dx

d tanx = sec2 x dx

d secx = secx tanx dx

Derivative of the Exponential Function

d e u = e u du dx dx

Example of derivative of Exponential Function

2

Differentiate: e x

2

Ans: 2x e x

Derivative of Ln

d (lnx) = 1/x dx

Definite Integral

b

a∫b f(x) dx= F(b)-F(a)= F(x)|a

a and b are the limits of integration.

If F(x)= ∫ f(x) dx

then

d F(x) = f(x) dx

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

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

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

To evaluate integrals of products of functions :

• Chain Rule

• Integration by parts

• Change of Variable Formula

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

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:

Example of Change of Variable Formula

Evaluate: 0∫1 2x (x2 + 1) 9 dx

Answ: 102.3

Integration by Parts

a∫b u(x) dv dx=

dx

b

= u(x) v(x)|a - a∫b v(x) du dx

dx

Integration by Parts

b

a∫b u v’ dx= u v|a - a∫b v u’ dx

Example of Integration by Parts

Compute 0∫π x sinx dx

Ans: π