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The concept of derivative of composite functions using the chain rule. It provides examples and formulas to help understand the product of derivatives evaluated appropriately for different functions. Students of calculus and mathematics can benefit from this document as study notes, summaries, or cheat sheets for understanding the chain rule.
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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
Try a Java applet.
The derivative of the composition of two non-constant functions is equal to the product
of their derivatives, evaluated appropriately.
We have the Chain Rule:
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.
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.
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
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
cos x .
We have f
′ (x) = e
cos x ( cos x)
′ = e
cos x ( − sin x) = − sin xe cos^ x
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