Differentiation of a function with a constant exponent

If the $n$ factors in the above result are each equal to $v$, we get

$$\frac{\frac{d}{dx}(v^n)}{v^n} = n\frac{\frac{dv}{dx}}{v}.$$

Therefore, $\frac{d}{dx}(v^n) = nv^{n-1}\frac{dv}{dx}$, (equation (VI) above).

When $v = x$ this becomes $\frac{d}{dx}(x^n) = nx^{n - 1}$ (equation (VIa) above).

We have so far proven equation (VI) only for the case when $n$ is a positive integer. In §5.15, however, it will be shown that this formula holds true for any value of $n$, and we shall make use of this general result now.

The derivative of a function with a constant exponent is equal to the product of the exponent, the function with the exponent diminished by unity, and the derivative of the function.