Extending the Power Rule to and With Theorem 3.3 and Exercise, we have shown that the Pow...

Extending the Power Rule to and With Theorem 3.3 and Exercise, we have shown that the Power Rule, , applies to any integer n. Later in the chapter, we extend this rule so that it applies to any rational number n.

a. Explain why the Power Rule is consistent with the formula .

b. Prove that the Power Rule holds for . (Hint: Use the definition of the derivative: .)

c. Prove that the Power Rule holds for .

d. Propose a formula for , for any positive integer n.

Exercise

Looking ahead: Power Rule for negative integers Suppose n is a negative integer and f(x) = xn. Use the following steps to prove that f′(a) = nan−1, which means the Power Rule for positive integers extends to all integers. This result is proved in Section 3.4 by a different method.

a. Assume that m = −n, so that m > 0. Use the definition .

Simplify using the factoring rule (which is valid for n > 0) xn − an = (x − a)(xn−1 + xn−2a + ⋯ + xan−2 + an−1) until it is possible to take the limit.

b. Use this result to find .