The Formula for
d r
dx x
When r Is a Rational Number
The goal here is to show that for any rational number r (Recall that a rational number is one that can be
written as the quotient of two integers.) and for any x > 0 the formula for the derivative of the function
d r
xr is
x = rxr¡1 : We begin with the special case of the reciprocal of a positive integer. Let n be a
dx
d n
positive integer and recall that we have already established the formula
x = nxn¡1 . Geometrically
dx
this means that the graph of y = xn has a tangent line at each point on its graph. Speci¯cally the slope of
the line tangent to the graph at the point (x; xn ) is nxn¡1 . In the ¯gure below locate the graph of y = x2
1
and the line tangent to the graph at ( 34 ; 49 ). The graph of the function f (x) = x n is obtain by re°ecting
the graph of y = xn in the line y = x. In the ¯gure below locate both the line y = x and the graph
1
resulting by re°ecting y = x2 in the line y = x. This graph is the graph of y = x 2 . It is geometrically
evident that the re°ection process takes lines to lines and in particular it takes tangent lines to tangent
lines. In the ¯gure below locate the re°ection in the line y = x of the tangent line located before. Its
1
re°ection is the line tangent to the graph of y = x 2 at the point ( 49 ; 34 ). In general when the line tangent
to the graph of y = xn at the point (x; xn ) is re°ected in the line y = x its re°ection is the line tangent
1
1
to the graph of y = x n at the point (x n ; x).
1
As a consequence the function f (x) = x n has a tangent line at each point of its graph except at the point
1
(0; 0). Recalling that having a tangent line is equivalent to being di®erentiable, we have established the
following proposition.
1
Proposition 1. The function f (x) = x n is di®erentiable at each x in its domain except for x = 0.
1
The Chain Rule will now be use to ¯nd the derivative of the function f (x) = x n : The domain
f is R
¡ of
1 ¢n
n
n
if n is odd and [0; 1) if n is even. Moreover for any x in the domain of f , we have f (x) = x
= x.
We already know that the outer function of this composition, xn , is di®erentiable. By Proposition 1 we
know that the inner function is also di®erentiable. So the Chain Rule can be applied to compute the
derivative of f at each x in the domain of f except, of course, at 0. By the Chain Rule,
d n
f (x) = nf n¡1 (x)f 0 (x):
dx
But because f n (x) = x. Consequently the derivative of the composition can also be computed without
d
d n
d n
the Chain Rule. We know
f (x) =
x = 1. Equating the two expressions for
f (x) we get that
dx
dx
dx
1
1
1 = nf n¡1 (x)f 0 (x) = n(x n )n¡1 f 0 (x) = nx1¡ n f 0 (x):
Solving this equation for f 0 (x) yields
f 0 (x) =
1 1 ¡1
d 1
1 1
xn
x n = x n ¡1 :
or
n
dx
n
d n
Note that this formula is the same as
x = nxn¡1 with n replaced by n1 .
dx
We now proceed to show that the same formula holds if n is replaced by any rational number. For any
rational number r there are two integers, m and n with n positive such that r = m
n . We will now use the
Chain Rule to prove the following assertion.
Theorem 1. Let r be a rational number and let x > 0. Then
d r
x = rxr¡1 :
dx
¡ 1 ¢m
m
Proof. First note that xr = x n = x n , a composition with outer function xm and inner function
1
x n . We know the derivative of the outer function and by Theorem 1, we also know the derivative of the
inner function. Thus by the Chain Rule
¡ 1 ¢m¡1 1 1 ¡1
d ¡ 1 ¢m
m m 1 1
d r
xn
= m xn
x =
x n = x n ¡ n + n ¡1 = rxr¡1 :
dx
dx
n
n