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Use the binomial theorem to prove the formula for the derivative of x^n

by RoRi

Consider the function f(x) = x^n for a positive integer n. Prove that

    \[ \frac{f(x+h) - f(x)}{h} = nx^{n-1} + \frac{n(n-1)}{2} x^{n-2} h + \cdots + nxh^{n-2} + h^{n-1}. \]

Conclude that f'(x) = nx^{n-1} by considering the limit of this as h \to 0.

Proof. We recall the binomial theorem,

    \[ (x+h)^n = \sum_{k=0}^n \binom{n}{k} x^k h^{n-k}. \]

So, then we calculate,

    \begin{align*} \frac{f(x+h) - f(x)}{h} &= \frac{1}{h} \left( (x+h)^n - x^n \right) \\ &= \frac{1}{h} \left( \left(\sum_{k=0}^n \binom{n}{k} x^k h^{n-k} \right) - x^n \right) \\ &= \frac{1}{h} \left( \sum_{k=0}^{n-1} \binom{n}{k} x^k h^{n-k} \right) \\ &= \sum_{k=0}^{n-1} \binom{n}{k} x^k h^{n-k-1} \\ &= nx^{n-1} + \frac{n(n-1)}{2}x^{n-2} h + \cdots + h^{n-1}. \qquad \blacksquare \end{align*}

Taking the limit as h \to 0 of both sides of the equation we conclude,

    \[ f'(x) = \lim_{h \to 0} \frac{f(x+h)-f(x)}{h} = nx^{n-1}. \]

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