Yet Another Calculus Text, 2007a

36 CHAPTER 1. DERIVATIVES
Hence
from which if follows that
n−1 dy
ny =1, (1.7.46)
dx
dy
dx = 1
ny n−1 = 1 n y1−n = 1 ( ) 1−n
x 1 1
n =
n
n x 1 n −1 , (1.7.47)
showing that the power rule works for rational powers of the form 1 n
. Note that
the above derivation is not complete since we began with the assumption that
y = x 1 n is differentiable. Although it is beyond the scope of this text, it may be
shown that this assumption is justified for x>0ifn is even, and for all x ≠0
if n is odd.
Now if m ≠ 0 is also an integer, we have, using the chain rule as above,
d
dx x m n
d
(
=
dx
= m
) m
x 1 n
(
x 1 n
) m−1 1
n x 1 n −1
= m n x m−1
n + 1 n −1
= m n x m n −1 . (1.7.48)
Hence we now see that the power rule holds for any non-zero rational exponent.
Theorem 1.7.10. If r ≠ 0 is any rational number, then
d
dx xr = rx r−1 . (1.7.49)
Example 1.7.18. With r = 1 2
in the previous theorem, we have
d √ 1 x =
dx 2 x− 1 1
2 =
2 √ x ,
in agreement with our earlier direct computation.
Example 1.7.19. If y = x 2 3 ,then
dy
dx = 2 3 x− 1 2
3 = .
3x 1 3
Note that dy
dx
is not defined at x = 0, in agreement with our earlier result showing
that y is not differentiable at 0.
Exercise 1.7.16. Find the derivative of f(x) =5x 4 5 .
We may now generalize 1.7.44 as follows: If u is a differentiable function of
x and r ≠ 0 is a rational number, then
d
dx ur = ru
r−1 du
dx . (1.7.50)