The Power Rule with Negative and Fractional Exponents

The power rule was probably first shown to you with whole numbers: the derivative of x³ is 3x², the derivative of x⁵ is 5x^4. But the rule itself never required the exponent to be a positive integer. It comes from the same limit definition of the derivative no matter what n is, so it applies just as cleanly when n is negative or a fraction. The only extra work is translation: you have to see that √x and 1/x² are powers of x in disguise before you can touch them with the rule.

Rewriting is mechanical once you know the two moves. A term in the denominator, like 1/x², becomes x raised to the negative of that exponent: x^−2. A root, like √x, becomes x raised to a fractional exponent: x^1/2; a cube root would be x^1/3. Once everything is a single power of x, bring the exponent down as a coefficient and subtract one from the exponent, exactly as before. Negative and fractional exponents stay negative and fractional after you differentiate — the rule changes the exponent's value by one, it does not change its sign or clear its denominator.

This topic is what turns the power rule from a trick for polynomials into a tool for almost any algebraic function. Once you can rewrite radicals and reciprocals as powers, the sum rule lets you differentiate term by term even when the terms look nothing alike at first glance. It also sets up later work: the chain rule will ask you to do this same rewriting step first, then differentiate the outside power, so getting comfortable with negative and fractional exponents now pays off immediately.