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Home Differentiation Derivatives of Rational Functions Theorem 7: Power Rule for Negative Exponents
 
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Theorem 7: Power Rule for Negative Exponents

THEOREM 7 (Power Rule for Negative Exponents)

Suppose u depends on x and n is a negative integer. Then for any value of x where du/dx exists and u ≠ 0, d(un)/dx exists and

02_differentiation-131.gif

PROOF

Since n is negative, n = -m where m is positive. Let y = un = u-m. Then y = 1/um. By the Lemma and the Power Rule,

02_differentiation-132.gif

02_differentiation-133.gif

The Quotient Rule together with the Constant, Sum, Product, and Power Rules make it easy to differentiate any rational function.

Power Rule for Negative Exponents

Home Differentiation Derivatives of Rational Functions Theorem 7: Power Rule for Negative Exponents

Last Update: 2006-11-05