The Power Rule for Derivatives (for rational powers)

Theorem:

If $f\,(x) = x^{\frac{p}{q}}$, where $p$ and $q$ are integers, then $f\,'(x) = \frac{p}{q} x^{\frac{p}{q}-1}$



Proof:

Suppose $$y = x^{\frac{p}{q}}$$ Raising both sides to the $q$ power, we get an equation involving integer powers of $x$ and $y$ $$y^q = x^p$$ As such, we can differentiate both sides using the Power Rule for Derivatives (for integer powers) to obtain $$qy^{q-1} \frac{dy}{dx} = px^{p-1}$$ Dividing both sides by $qy^{q-1}$ to solve for $\frac{dy}{dx}$, and massaging the result yields $$\begin{array}{rcl} \frac{dy}{dx} &=& \frac{px^{p-1}}{qy^{q-1}}\\\\ &=& \frac{p}{q} x^{p-1} y^{1-q} \end{array}$$ Lastly, we recall that $y = x^{\frac{p}{q}}$, and substitute this expression to remove the $y$ above $$\begin{array}{rcl} \frac{dy}{dx} &=& \frac{p}{q} x^{p-1} (x^{\frac{p}{q}})^{1-q}\\\\ &=& \frac{p}{q} x^{p-1} x^{\frac{p}{q} - p}\\\\ &=& \frac{p}{q} x^{\frac{p}{q} - 1} \end{array}$$

Thus, if $f\,(x) = x^{\frac{p}{q}}$, where $p$ and $q$ are integers, it must be true that $f\,'(x) = \frac{p}{q} x^{\frac{p}{q}-1}$

QED.