Math Wiki

Elementary power rule of derivatives/Proof

1,078pages on
this wiki
Add New Page
Talk0 Share

\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 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)


\frac{d}{dx} x^n= \frac{d}{dx} (e^{\ln(x)})^n
\frac{d}{dx} x^n= \frac{d}{dx} e^{n \cdot\ln(x)}
\frac{d}{dx} x^n= e^{n \cdot\ln(x)} \cdot \frac{d}{dx}(n \cdot \ln(x))
\frac{d}{dx} x^n= x^{n} \cdot n \cdot \frac{1}{x}
\frac{d}{dx} x^n= n\cdot x^{n-1}

Proof by Binomial Expansion

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


  • The Binomial Expansion:
(a+b)^n = \sum_{i=0}^n \frac{n!}{i!(n-i)!} \cdot a^{n-i} \cdot b^i
  • The first principle limit definition of the derivative
\frac{d}{dx} f(x) = \lim_{h\to 0} \frac{f(x+h) - f(x)}{h}


\frac{d}{dx} x^n = \lim_{h\to 0} \frac{(x+h)^n - x^n}{h}
\frac{d}{dx} x^n = \lim_{h\to 0} \frac{\left(\sum_{i=0}^n \frac{n!}{i!(n-i)!} \cdot x^{n-i} \cdot h^i\right) - x^n}{h}
\frac{d}{dx} x^n = \lim_{h\to 0} \frac{\left(x^n + nx^{n-1}h + \sum_{i=2}^n \frac{n!}{i!(n-i)!} \cdot x^{n-i} \cdot h^i\right) - x^n}{h}
\frac{d}{dx} x^n = \lim_{h\to 0} \frac{nx^{n-1}h + \sum_{i=2}^n \frac{n!}{i!(n-i)!} \cdot x^{n-i} \cdot h^i}{h}
\frac{d}{dx} x^n = \lim_{h\to 0} nx^{n-1} + \sum_{i=2}^n \frac{n!}{i!(n-i)!} \cdot x^{n-i} \cdot h^{i-1}
\frac{d}{dx} x^n = nx^{n-1} + \lim_{h\to 0} \sum_{i=2}^n \frac{n!}{i!(n-i)!} \cdot x^{n-i} \cdot h^{i-1}
\frac{d}{dx} x^n = nx^{n-1} + 0
\frac{d}{dx} x^n = nx^{n-1}

Ad blocker interference detected!

Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.

Also on Fandom

Random Wiki