Code Scholars Tutoring · AP Calculus AB

Product and quotient rules

Unit 2 · Topics 2.8, 2.9. Original AP Calculus AB notes, worked example, and self-check.

Understand the idea

A product derivative differentiates one factor at a time and adds the two terms. For a quotient, multiply the denominator by the numerator derivative, subtract numerator times denominator derivative, and divide by the denominator squared. Simplification before differentiating may shorten the work. A derivative formula is valid only where the original function and relevant derivatives exist.

(uv)'=u' v+uv',\qquad\left(\frac uv\right)'=\frac{vu'-uv'}{v^2}

Worked example

Differentiate $\displaystyle f(x)=x^2e^x$ and evaluate $\displaystyle f'(1)$ .

The product rule gives

\displaystyle f'(x)=2xe^x+x^2e^x=e^x(2x+x^2)

. At

\displaystyle 1

, this equals

\displaystyle 3e

. Differentiating both factors and multiplying would incorrectly give

\displaystyle 2xe^x

Try it yourself

Find the derivative of $\displaystyle (x+1)/(x-1)$ .

Show solution and reasoning

The quotient rule gives $\displaystyle ((x-1)-(x+1))/(x-1)^2=-2/(x-1)^2$ , for $\displaystyle x\ne1$ .

Unit 2 MCQ