Concept preview

What you will be able to do

1. 01State the quotient rule and reproduce it correctly, including the exact order of the numerator terms, the subtraction, and the squared denominator.
2. 02Differentiate a rational function of the form $f/g$ and evaluate the resulting derivative at a specified point.
3. 03Decide when a quotient is differentiated most efficiently by the quotient rule and when rewriting it as a product with a negative power is faster and less error-prone.

You already differentiate sums, differences, products, and compositions. The last elementary combination is the quotient — one function divided by another. Rational functions like $\dfrac{2x+1}{x-3}$, the tangent function written as $\dfrac{\sin x}{\cos x}$, and average-cost expressions like $\dfrac{C(q)}{q}$ are all quotients, and each shows up on the AP exam. The rule that differentiates them is short to write but easy to misremember, so we will build it, drill its exact form, and practice choosing when not to use it.