Calculus- Early Transcendentals, 2021a

138 Derivatives
In practice, of course, you will need to use more than one of the rules we have developed to compute
the derivative of a complicated function.
♣
Example 4.32: Derivative of Quotient
Compute the derivative of
f (x)= x2 − 1
x √ x 2 + 1 .
Solution. The “last” operation here is division, so to get started we need to use the quotient rule first. This
gives
f ′ (x) = (x2 − 1) ′ x √ x 2 + 1 − (x 2 − 1)(x √ x 2 + 1) ′
x 2 (x 2 + 1)
= 2x2√ x 2 + 1 − (x 2 − 1)(x √ x 2 + 1) ′
x 2 (x 2 .
+ 1)
√
Now we need to compute the derivative of x x 2 + 1. This is a product, so we use the product rule:
Finally, we use the chain rule:
d
dx
And putting it all together:
d
dx x √x 2 + 1 = x d dx
√
x 2 + 1 +
√
x 2 + 1.
√
x 2 + 1 = d dx (x2 + 1) 1/2 = 1 2 (x2 + 1) −1/2 (2x)=
x
√
x 2 + 1 .
f ′ (x) = 2x2√ x 2 + 1 − (x 2 − 1)(x √ x 2 + 1) ′
x 2 (x 2 .
+ 1)
(
2x 2√ x 2 + 1 − (x 2 x
− 1) x√ x
=
2 + 1 + √ )
x 2 + 1
x 2 (x 2 .
+ 1)
This can be simplified of course, but we have done all the calculus, so that only algebra is left.
Example 4.33: Chain of Composition
√
√
Compute the derivative of 1 + 1 + √ x.
♣
Solution. Here we have a more complicated chain of √compositions, so we use the chain rule twice. At
the outermost “layer” we have the function g(x)=1 + 1 + √ x plugged into f (x)= √ x, so applying the
chain rule once gives
√
√
d
1 + 1 + √ x = 1 ( √
1 + 1 + √ ) −1/2 ( √
d
x 1 + 1 + √ )
x .
dx
2
dx