Let the secant become a tangent.

An average rate is the slope between two points: [f(x + h) − $f(x)$]/h. Let the spacing h approach zero to obtain the tangent gradient $f'(x)$. The derivative is itself a function, so its value varies with x.

Expand before cancelling h, then take the limit. Substituting h = 0 too early creates $\frac{0}{0}$. For a positive integer n, the resulting power rule is d($x^{n}$)/dx = nxⁿ⁻¹; a constant differentiates to zero.