Power rule
Proof by implicit differentiation

A more straightforward generalization of the power rule to rational exponents makes use of implicit differentiation.

Let y = x r = x p / q {\displaystyle y=x^{r}=x^{p/q}}, where p, q ∈ Z {\displaystyle p,q\in \mathbb {Z} } so that r ∈ Q {\displaystyle r\in \mathbb {Q} }.

Then,

Differentiating both sides of the equation with respect to x {\displaystyle x},

Solving for d y d x {\displaystyle {\frac {dy}{dx}}},

Since y = x p / q {\displaystyle y=x^{p/q}},

Applying laws of exponents,

Thus, letting r = p q {\displaystyle r={\frac {p}{q}}}, we can conclude that d d x x r = r x r − 1 {\displaystyle {\frac {d}{dx}}x^{r}=rx^{r-1}} when r {\displaystyle r} is a rational number.
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