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Use implicit differentiation to derive the formula y’ = rx^(r-1) for rational r

by RoRi

Let y = x^r for r = \frac{m}{n} for integers m and n. This implies

    \[ y = x^r = x^{\frac{m}{n}} \quad \implies \quad y^n = x^m. \]

If we assume that the derivative y' exists then derive the formula

    \[ y' = rx^{r-1} \]

using the formula for the derivative of an integer power and implicit differentiation.

Starting with the equation y^n = x^m we differentiate each side with respect to x (where y is a function of x so we need to use the chain rule).

    \begin{align*} &&y^n &= x^m \\ \implies && ny^{n-1} y' &= mx^{m-1} &(\text{Using derivative for integer powers})\\ \implies && y' &= \frac{mx^{m-1}}{ny^{n-1}} \\ &&&= \left( \frac{m}{n} \right) \left( \frac{x^{m-1}}{(x^r)^{n-1}} \right) \\ &&&= \left( \frac{m}{n} \right) (x^{m-1-m+r})\\ &&&= r x^{r-1}. \end{align*}

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