Chapter 3 | Derivatives
295
Example 3.57 Using the Chain Rule with Functional Values
Let h ( x ) = f ⎛
⎝ g ( x ) ⎞ ⎠ . If g (1) =4, g ′(1) =3, and f ′(4) =7, find h ′(1).
Solution Use the chain rule, then substitute.
h ′(1) = f ′ ⎛
⎞ ⎠ g ′(1) Apply the chain rule.
⎝ g (1)
Substitute g (1) =4and g ′(1) =3.
= f ′(4) ·3
=7·3
Substitute f ′(4) =7.
=21
Simplify.
Given h ( x ) = f ⎛
⎝ g ( x ) ⎞ ⎠ . If g (2) =−3, g ′(2) =4, and f ′(−3) =7, find h ′(2).
3.40
The Chain Rule Using Leibniz’s Notation As with other derivatives that we have seen, we can express the chain rule using Leibniz’s notation. This notation for the chain rule is used heavily in physics applications. For h ( x ) = f ⎛ ⎝ g ( x ) ⎞ ⎠ , let u = g ( x ) and y = h ( x ) = f ( u ). Thus, h ′( x ) = dy dx , f ′ ⎛ ⎝ g ( x ) ⎞ ⎠ = f ′( u ) = dy du and g ′( x ) = du dx . Consequently, dy dx = h ′( x ) = f ′ ⎛ ⎝ g ( x ) ⎞ ⎠ g ′( x ) = dy du · du dx .
Rule: Chain Rule Using Leibniz’s Notation If y is a function of u , and u is a function of x , then dy dx = dy du · du dx .
Example 3.58 Taking a Derivative Using Leibniz’s Notation, Example 1
Find the derivative of y = ⎛
⎞ ⎠
5
⎝ x
.
3 x +2
Solution First, let u =
dy du .
Thus, y = u 5 . Next, find du dx
x 3 x +2 .
and
Using the quotient rule,
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