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Derivatives of Composite Functions
Recall that the composite function or composition of two functions is the function
obtained by applying them one after the other.
For example, If f (x) =
x
and g(x) = x^3 + 2, then
f (g(x)) =
g(x)
x^3 + 2
and g(f (x)) = (^) (f (x))
3
x
x^3
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The derivative of the composition of two non-constant functions is equal to the product
of their derivatives, evaluated appropriately.
The Chain Rule
We have the Chain Rule:
g(h(x))
= g
(h(x))h
(x)
Example 1: Using g(x) =
x
= x
− 1 and h(x) = x^3 + 2,
we have g ′ (x) = ( − 1 )x −^2 and h ′ (x) = 3 x^2 , g ′ (h(x)) = ( − 1 )(h(x)) −^2 , so we get
x^3 + 2
= g
′ (h(x))h
′ (x) = ( − 1 )(h(x))
− 2 ( 3 x
2 ) =
( − 1 )(x
3
− 2 ( 3 x
2 ) =
− 3 x^2
(x^3 + 2 )^2
On the other hand,
x^3
h(g(x))
= h
′ (g(x))g
′ (x) =
3 (g(x))
2 ( − x
− 2 ) = 3 (x
− 1 )
2 ( − x
− 2 ) = − 3 x
− 4 , as expected.
Example 2: Let g(x) = x^3 , and h(x) = x^2 , so that
g(h(x)) = h(x)^3 = (x^2 )^3 = x^6.
Then g ′ (x) = 3 x^2 , so g ′ (h(x)) = 3 (h(x))^2 , and h ′ (x) = 2 x ,
so the Chain Rule gives us
g ′ (h(x))
= g ′ (h(x))h ′ (x) =
3 (h(x))^2
( 2 x) =
3 (x^2 )^2
( 2 x) =
3 x^4
( 2 x) = 6 x^5 , as expected.
Example 3: Let g(x) = x^3 + 3, and h(x) = x^2 + 2, so that
g(h(x)) = (h(x))^3 + 3 = (x^2 + 2 )^3 + 3.
Then g ′ (x) = 3 x^2 , so g ′ (h(x)) = 3 (h(x))^2 , and h ′ (x) = 2 x ,
so the Chain Rule gives us
g ′ (h(x))
= g ′ (h(x))h ′ (x) =
3 (h(x))^2
( 2 x) =
3 (x^2 + 2 )^2
( 2 x) =
6 x(x
2
2
Example 4: Find f ′ (x) if f (x) =
x^4 + x^2 + 1.
We let g(x) = x
1 (^3) and h(x) = x^4 + x^2 + 1 so that f (x) = g(h(x)).
Then g ′ (x) =
1 3 x
− (^23) , g ′ (h(x)) = 1 3
(h(x))
− (^23) , and h ′ (x) = 4 x^3 + 2 x ,
so we have f ′ (x) = g ′ (h(x))h ′ (x) =
(h(x))
− (^23) ( 4 x^3 + 2 x) =
2 x( 2 x
2
3 (x^4 + x^2 + 1 )
2 3
Example 8: Find f ′ (x) if f (x) = e
cos x .
We have f
′ (x) = e
cos x ( cos x)
′ = e
cos x ( − sin x) = − sin xe cos^ x
Example 9: Find f ′ (x) if f (x) = sin
e
tan x ) .
We have f
′ (x) = cos
e
tan x ) ( e
tan x )′^ = cos
e
tan x )^ e
tan x ( tan x)
′
cos
e
tan x )^ e
tan x sec
2 x
