\frac{d}{dx}(x^n)=nx^{n-1} , for n is a natural number.

Proof by Induction


  • The product rule of derivatives, \frac{d}{dx}(uv_=uv'+vu'


The proof proceeds by mathematical induction. Take the base case k=0. Then:


The induction hypothesis is that the rule is true for n=k:


We must now show that it is true for n=k+1:


Since the power rule is true for k=0 and given k is true, k+1 follows, the power rule is true for any natural number.


Proof by Exponentiation

This proof is validates the power rule for all real numbers n such that the derivative \frac{d}{dx}(x^n)=nx^{n-1}


  • The exponential rule of derivatives, \frac{d}{dx}(e^x)=e^x
  • The chain rule of derivatives, \frac{d}{dx}f(g(x))=f'(g(x))\cdot g'(x)


&=x^n\cdot n\cdot\frac1x\\

Proof by Binomial Expansion

This proof is only valid for positive real integer n exponents, x^n


  • The Binomial Expansion:
(a+b)^n=\sum_{k=0}^n\frac{n!}{k!(n-k)!}\cdot a^{n-k}\cdot b^k
  • The first principle limit definition of the derivative


&=\lim_{h\to0}\frac{\displaystyle\left(\sum_{k=0}^n\frac{n!}{k!(n-k)!}\cdot x^{n-k}\cdot h^k\right)-x^n}{h}\\
&=\lim_{h\to0}\frac{\displaystyle\left(x^n+nx^{n-1}h+\sum_{k=2}^n\frac{n!}{k!(n-k)!}\cdot x^{n-k}\cdot h^k\right)-x^n}{h}\\
&=\lim_{h\to0}\frac{\displaystyle nx^{n-1}h+\sum_{k=2}^n\frac{n!}{k!(n-k)!}\cdot x^{n-k}\cdot h^k}{h}\\
&=\lim_{h\to0}\left[nx^{n-1}+\sum_{k=2}^n\frac{n!}{k!(n-k)!}\cdot x^{n-k}\cdot h^{k-1}\right]\\
&=nx^{n-1}+\lim_{h\to0}\sum_{k=2}^n\frac{n!}{k!(n-k)!}\cdot x^{n-k}\cdot h^{k-1}\\

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