The Power Rule

Prerequisites:

The power rule states that

ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}

To take a simple example consider x1x^1, according to this law the derivative is 1x0=11x^0 = 1 which it is! To see why the power rule is true consider the limit definition of a derivative

lim⁡h→0(x+h)n−xnh\lim_{h \to 0} \frac{(x+h)^n - x^n}{h}

We can expand (x+h)n(x+h)^n using the binomial theorem

lim⁡h→0(xn+nxn−1h+h2(… ))−xnh=lim⁡h→0nxn−1+h(… )\lim_{h \to 0} \frac{(x^n + nx^{n-1}h + h^2(\dots)) - x^n}{h} = \lim_{h \to 0} nx^{n-1} + h(\dots)

Therefor the derivative is nxn−1nx^{n-1}