The power rule poses an exceptional opportunity for students to explore a topic beginning with inductive pattern finding rather than deductive proof. I often hear math teachers talking about how finding patterns is an essential part of math. Which it is! But how often do we model this for our students? All too often, teachers present a formula (like the power rule) before giving students a chance to think and spot patterns for themselves. If we want students to see that spotting patterns is essential to mathematics, then this needs to be a core part of our teaching practice.
After doing some work with numerical derivatives, I introduce calculus with limits by showing my students this picture and asking them to approximate the slope of this graph at x=3.
