Download Notes 2.4

Section 2.4: The Chain Rule
The chain rule deals with composite functions. What is a composite function??
Derivative can be found without the Chain Rule
Composite Function: Use the Chain Rule
y = x2 + 1
y = √x2 + 1
y = sin x
y = sin 6x
y = 3x + 2
y = (3x + 2)5
y = x + tan x
y = x + tan x2
The Chain Rule
dy ___
dy
du
___
___
=
dx
dx
du
d
or ___ = f '(g(x)) g'(x)
dx
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I will not show u­substitution for the chain rule, because it is not necessary. We will use u­substitution in Chapter 4 for Integration.
Ex. 1: FInd the derivative of y = (x2+1)3
The inside function is (x2+1). (g(x))
The outside function is x3. (f(x))
d
___
= f '(g(x)) g'(x)
dx
(Derivative of the outside function)x(derivative of the inside function) y'= 3(x2+1)2 (2x) = 6x((x2+1)2
Derivative of the inside function (x2+1)
Derivative of the outside function x3
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You do: f(x) = (3x – 2x2)3
f' (x) = (Derivative of Outside function) x (Derivative of the inside function).
f' (x) =
Ex. 2: f(x) = ∛(x2-1)2
f'(x) = (DOF)(DIF)
f'(x) =
3
Ex. 3: f(x) = –7
___________
(2t – 3)2
Rewrite: f(x) = f'(x) =
4
Ex 4: f(x) = x2 √1-x2
5
x
__________
Ex 5: f(x) = 2
∛x +4
f'(x)=
6
x2
3x­1
Ex. 6: y = _______
(
x2+3
)
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Trigonometric Functions with the Chain Rule
y' = Ex. 10: a. y = sin 2x
b. y = cos(x­1)
c. y = tan 3x
y' = y' = Parentheses and Trigonometric Functions
Ex. 11: a. y = cos 3x2 = cos (3x2)
b. y = (cos 2) x2
c. y = cos (3x)2
d. y = cos2x = (cos x)2
e. y = √cos x = (cos x)1/2
Ex. 12: f(t) = sin34t
Note: We will use the chain rule more than once!!
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