Non-rigorous Proof of the Power Rule of Derivatives

 

Non-rigorous Proof of the Power Rule of Derivatives

By Clement Charles

 Using the binomial expansion to prove the power rule, which is the generalized rule for derivatives of powers (including polynomials).

Consider the function:

Where .

Recall the definition of the derivative is:

Therefore, the derivative of the function is:

By the binomial expansioncan be expressed as:

In summation notation:

Therefore, the limit becomes:

Factoring an h:

Recast this equation:

NB:

For:

Since terms for k > 1 are 0, it means that the expression can be reduced to:

NB:

And:

So that: