The Power Rule: An Infinitesimal Approach

This is the derivation you probably didn’t get in Calculus I

This derivation of the power rule does not use limits. Its inspiration is Abraham Robinson’s non-standard analysis and hyper-real numbers.

If you learned a derivation of the power rule, it probably started something with a definition of the derivative:

We’ll take a fresh approach using infinitesimals. You can decide which way you prefer.

Above we see a generic power function of degree n. The coefficient is 1, to keep things tidy. You’ll be able to extend what follows to include any coefficient.

What is the slope of the function at the indicated point on the left-hand side of the diagram?

The right-hand side of the diagram shows this point at infinite magnification. We also see a neighbouring point which is infinitely close to our point of interest. The slope of the line between this to points is given below.

We’ll continue forward with two specific cases — n=2 and n=3. Look for a pattern from which we can generalize.