110 3. LIMITS, DERIVATIVES, RULES, AND THE MEANING OF THE DERIVATIVE
Solution:
Apply the chain rule to the functional composition for which the outer function is f .x/ D e
x
,
and the inner function is g.x/ D 2x. For these, f
0
.x/ D e
x
and g
0
.x/ D 2. So:
h
0
.x/ D e
2x
2
D 2e
2x
˙
Example 3.64 Compute the derivative of q.x/ D sin
x
2
.
Solution:
Apply the chain rule to the functional composition for which the outer function is f .x/ D
sin.x/, and the inner function is g.x/ D x
2
. For these, f
0
.x/ D cos.x/ and g
0
.x/ D 2x. So:
q
0
.x/ D cos
x
2
2x
D 2x cos
x
2
˙
e chain rule avoids a whole lot of multiplying out in some cases. Technically, we could do the
following example without the chain rule, but it would be purely awful.
Example 3.65 Compute the derivative of r.x/ D
x
2
C x C 1
7
.
Solution:
Apply the chain rule to the functional composition for which the outer function is f .x/ D x
7
,
and the inner function is g.x/ D x
2
C x C 1.
For these, f
0
.x/ D 7x
6
and g
0
.x/ D 2x C 1.
So:
r
0
.x/ D 7
x
2
C x C 1
6
.
2x C 1
/