Calculus: Analytic Geometry and Calculus, with Vectors

3.5 Difference quotients and derivatives 157

have y = x and must prove that

This is true because if y = x, then Ay = Ox, so Ay/Ax = 1 and dy/dx = 1.
The case n = 2 is covered in the application under the four-step rule

headline. There are several somewhat different ways to obtain (3.563)
for greater integer values of n. Perhaps the most informative method
consists of using the product formula (3.565) to obtain

d u1u2u3__d(ulu2)u3=d UIU2 d U3

dx dx dx u3 + UIU2 dx

= dxl U2U3 + u1 d U3 + UIU2 d

and then putting ul = U2 = u3 = x to obtain

(3.572)

dxn= nxn-1

dx

when n = 3. Another application of the same idea, in which the product
u1u2u3u4 is written as the product (ulu2u3)u4 of two factors, gives

du1u2u3u4 dul due du3 duo

dx dx U2U3U4 + uldxU3U4 + ulu2 ax, U4 + UIU2U3dx

and putting n1 = u2 = u3 = U4 = x gives the formula (3.572) for the
case in which n = 4. The same procedure gives the result for greater
integers. Perhaps the simplest proof can be based upon the fact that if

(3.572) holds for a given n, then use of the product rule gives

dx'a+1 d

dx dx

x.x" = xnx"-1 + x".1 = (n + l)xn.

7

Since the formula is valid when n = 0, mathematical induction shows that

it is valid when n is a nonnegative integer. In case n is a negative integer

(so that -n is a positive integer) and x 0 0, the result is proved by the

calculation

dxn d (^11)

dx = dx x-n - X-2-

whichinvolves the formula for the derivative of a quotient. In case n is
a constant which is not an integer, (3.563) is still valid at least when
x > 0. Proof of this appears in Theorem 9.27, and proof of (3.564) then

follows from the chain rule of Theorem 3.65.