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

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)

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

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'

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

You Try: Find the derivative of .You Try: Find the derivative of .f (x)=(5x2 +2x5 )7

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

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

C. differentiating quotients with constant numeratorsC. differentiating quotients with constant numerators

Ex. 4: Differentiate .Ex. 4: Differentiate .f (x)=−5

(5x−4)3

You Try: Differentiate .You Try: Differentiate .g(t)=−3

(4x+6)2

III. Simplifying derivativesIII. Simplifying derivatives

Ex. 5: Find the derivative of .Ex. 5: Find the derivative of .f (x)=x3 2−3x2

You Try: Find the derivative of .You Try: Find the derivative of .f (x)=x2 5−2x2

Ex. 6: Find the derivative of .Ex. 6: Find the derivative of .f (x)=2x

5x2 −33

You Try: Find the derivative of .You Try: Find the derivative of .f (x)=x

x2 −63

Ex. 7: Find the derivative of .Ex. 7: Find the derivative of .f (x)=4x−9x2 +2

⎛⎝⎜

⎞⎠⎟

2

You Try: Find the derivative of .You Try: Find the derivative of .f (x)=2x+63x2 +1

⎛⎝⎜

⎞⎠⎟

2

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

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

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

You Try: Find the derivative of .You Try: Find the derivative of .f (t)=sin2(6t−3)

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( )