# Blog

## How to prove the product rule with limits? ## What is the product rule for derivatives?

• Proving the product rule for derivatives. The product rule tells us how to find the derivative of the product of two functions: The AP Calculus course doesn't require knowing the proof of this rule, but we believe that as long as a proof is accessible, there's always something to learn from it.

## What is the key argument in the proof of product rule?

• The key argumenthere is the next to last line, where we have used the fact that both fand gare differentiable, hence the limit can be distributed across the sum to give the desired equality. Title proof of product rule Canonical name ProofOfProductRule

## What are the applications of the product rule?

• Among the applications of the product rule is a proof that when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0.  