Concept preview

What you will be able to do

1. 01State and apply the product rule $(fg)' = f'g + fg'$ to differentiate products of functions, including polynomial×trig and polynomial×exponential combinations.
2. 02Explain, using a counterexample, why $(fg)'=f'g'$ is not a valid differentiation rule in general.
3. 03Use the product rule to evaluate a derivative at a specific point and to find the equation of the tangent line to a product of functions.

Whenever a function is a product of two simpler functions multiplied together — a polynomial times a sine, a power times an exponential, any $f(x)\cdot g(x)$ that you cannot easily expand — you need a rule for its derivative. The natural first guess is that the derivative of a product should be the product of the derivatives, just as the derivative of a sum is the sum of the derivatives. That guess feels reasonable and it is completely false. Seeing exactly why it fails is the fastest route to trusting the correct rule.