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Elementary power rule of derivatives/Proof

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

Proof by Induction

Prerequisites

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

Proof

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

\frac{d}{dx}x^0=\frac{d}{dx}1=0=0x^{-1}

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

\frac{d}{dx}(x^k)=kx^{k-1}

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

\begin{align}
\frac{d}{dx}(x^{k+1})&=\frac{d}{dx}xx^k\\
&=x\frac{d}{dx}x^k+x^k\\
&=xkx^{k-1}+x^k=kx^k+x^k\\
&=(k+1)x^k
\end{align}

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.

QED

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}

Prerequisites

  • 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)

Proof

\begin{align}
\frac{d}{dx}(x^n)&=\frac{d}{dx}\left(e^{\ln(x)}\right)^n\\
&=\frac{d}{dx}\left(e^{n\ln(x)}\right)\\
&=e^{n\ln(x)}\cdot\frac{d}{dx}\big(n\ln(x)\big)\\
&=x^n\cdot n\cdot\frac1x\\
&=nx^{n-1}
\end{align}

Proof by Binomial Expansion

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

Prerequisites

  • 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
\frac{d}{dx}f(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

Proof

\begin{align}
\frac{d}{dx}(x^n)&=\lim_{h\to0}\frac{(x+h)^n-x^n}{h}\\
&=\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}\\
&=nx^{n-1}+0\\
&=nx^{n-1}
\end{align}

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