# Thread: Proof by induction for power rule in derivatives

1. ## Proof by induction for power rule in derivatives

I'm looking for a proof by induction for the power rule of derivatives, i.e. $\displaystyle \frac{d}{dx}x^n=nx^{n-1}$ for $\displaystyle n\geq1$

In the proof, I need to use the following facts:

$\displaystyle x^{n+1}=xx^n$
and
$\displaystyle g'(x)=x(f'(x))+f(x)$

Can someone help out here?

2. Originally Posted by superphysics
I'm looking for a proof by induction for the power rule of derivatives, i.e. $\displaystyle \frac{d}{dx}x^n=nx^{n-1}$ for $\displaystyle n\geq1$

In the proof, I need to use the following facts:

$\displaystyle x^{n+1}=xx^n$
and
$\displaystyle g'(x)=x(f'(x))+f(x)$

Can someone help out here?
Base case:

$\displaystyle \frac{d}{dx}x^1 = \frac{d}{dx}x = 1$

Prove this directly from the definition of the derivative.

Suppose for some $\displaystyle k>0\ k \in \mathbb{N}$ that

$\displaystyle \frac{d}{dx}x^k=kx^{k-1}$

Then consider:

$\displaystyle \frac{d}{dx}x^{k+1}= \frac{d}{dx}[x \ x^k]=x^k + x\ (kx^{k-1})=(k+1)x^k$

The second from last step uses the product rule the assumption and the base case.

From there it is routine to complete the proof

RonL