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Chapter 3 | Derivatives
⎛ ⎝ x 3
⎞ ⎠ −sin
⎛ ⎝ a 3
⎞ ⎠
⎛ ⎝ a 3
⎞ ⎠
sin
sin u −sin
lim x → a
= lim
x 3 − a 3
u − a 3
u → a 3
= d
du (sin
u ) u
= a 3
⎛ ⎝ a 3 ).
=cos
Thus, h ′( a ) =cos ⎛ ⎞ ⎠ ·3 a 2 . In other words, if h ( x ) = sin ⎛ ⎝ x 3 ⎞ ⎝ a 3
⎠ , then h ′( x ) =cos ⎛
⎞ ⎠ ·3 x 2 . Thus, if we think of h ( x ) = sin ⎛ ⎝ x 3 ⎞
⎝ x 3
⎠ as the composition
⎞ ⎠ where f ( x ) = sin x and g ( x ) = x 3 , then the derivative of h ( x ) = sin ⎛ ⎝ x 3 ⎞
⎛ ⎝ f ∘ g
⎞ ⎠ ( x ) = f
⎛ ⎝ g ( x )
⎠ is the product of the derivative of g ( x ) = x 3 and the derivative of the function f ( x ) = sin x evaluated at the function g ( x ) = x 3 . At this point, we anticipate that for h ( x ) = sin ⎛ ⎝ g ( x ) ⎞ ⎠ , it is quite likely that h ′( x ) =cos( g ( x )) g ′( x ). As we determined above, this is the case for h ( x ) = sin ⎛ ⎝ x 3 ⎞ ⎠ . Now that we have derived a special case of the chain rule, we state the general case and then apply it in a general form to other composite functions. An informal proof is provided at the end of the section. Rule: The Chain Rule Let f and g be functions. For all x in the domain of g for which g is differentiable at x and f is differentiable at g ( x ), the derivative of the composite function h ( x ) = ⎛ ⎝ f ∘ g ⎞ ⎠ ( x ) = f ⎛ ⎝ g ( x ) ⎞ ⎠
is given by
h ′( x ) = f ′ ⎛ ⎞ ⎠ g ′( x ). Alternatively, if y is a function of u , and u is a function of x , then dy dx = dy du · du dx . ⎝ g ( x )
(3.17)
Watch an animation (http://www.openstax.org/l/20_chainrule2) of the chain rule.
Problem-Solving Strategy: Applying the Chain Rule 1. To differentiate h ( x ) = f ⎛ ⎝ g ( x ) ⎞ 2. Find f ′( x ) and evaluate it at g ( x ) to obtain f ′ ⎛ ⎝ g ( x ) ⎞ ⎠ . 3. Find g ′( x ). 4. Write h ′( x ) = f ′ ⎛ ⎝ g ( x ) ⎞ ⎠ · g ′( x ).
⎠ , begin by identifying f ( x ) and g ( x ).
Note : When applying the chain rule to the composition of two or more functions, keep in mind that we work our way from the outside function in. It is also useful to remember that the derivative of the composition of two functions can be thought of as having two parts; the derivative of the composition of three functions has three parts; and so on. Also, remember that we never evaluate a derivative at a derivative.
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