#Product rule calculus full#

In fact, I haven’t motivated them at all, but I don’t need to explain my clever methods to you. I will perform some algebraic manipulations here in blue - they may appear unmotivated to you, but that’s your fault.

This proof crucially involves cleverness, but since you’re not clever, you never would have thought of it yourself. In other words, reader, I am clever and you are not. The author even admits that the proof is unsatisfying and unedifying and apologizes in advance for its opaque maneuvers! Some proofs involve “clever steps that may appear unmotivated to a reader”. The author merely proves the theorem, dryly and without understanding or purpose. There is no motivating example and no geometric intuition is called upon.

#Product rule calculus full#

I understand that textbooks have limited space and are no substitute for a full curriculum, but I think we can all agree that this is awful. Should we try to motivate the entire discussion with a particularly intuitive pair of functions whose product has some real-world significance? Should we interpret the product of two functions geometrically, as the area of the corresponding rectangle? If properly motivated and explained, do we actually gain anything by doing the rigorous proof via limits? The Status QuoĪs a foil, here is the introduction to and proof of the product rule from the textbook that I teach out of. Should we try to let the students discover the formula on their own? Should we perhaps lead them into a trap by suggesting that the derivative of a product of two functions is the product of the derivatives and let them find counterexamples? Should we state the theorem, but let the students try to prove it on their own? Should we perhaps have an entire mini-lesson on what it even means to have a product of two functions? Some teachers might simply write the rule on the board, expect students to accept it, and immediately launch into examples. How a calculus teacher chooses to do this probably says a lot about their pedagogy and educational priorities. At some point in every calculus class, we must discover and prove the product rule for derivatives.