The Product Rule: Interactive Slope Builder

The product rule says the derivative of f(x) times g(x) equals f-prime(x) times g(x) plus f(x) times g-prime(x): differentiate the first factor times the second, plus the first factor times the derivative of the second.

When you have two functions in parentheses multiplied together, you cannot simply take the derivative of both of them at the same time and multiply them. Instead, you must use the Product Rule.

To make the Product Rule simple, think of the first function as term A and the second function as term B:

1. Differentiate the first, keep the second

• Take the derivative of term A (giving you A') and leave term B alone.

2. Keep the first, differentiate the second

• Leave term A alone and multiply it by the derivative of term B (giving you B').

3. Add them together

• Combine both parts to get your final derivative: A'B + AB'.

What is the product rule for derivatives?

\begin{gathered} \underbrace{\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)}_{\text{Textbook Formula}} \\\\ \underbrace{(A \cdot B)' = {\color{#34d399}A'}B + A{\color{#eab308}B'}}_{\text{A} \cdot \text{B Version}} \end{gathered}

Product rule worked example: x squared times sine x

Find the derivative of h(x) = x² sin(x).

1. Identify the two functions: f(x) = x² and g(x) = sin(x).

2. Compute their individual derivatives:

f'(x) = 2x \quad \text{and} \quad g'(x) = \cos(x)

3. Set up the Product Rule formula:

h'(x) = (2x)(\sin x) + (x^2)(\cos x) = 2x\sin x + x^2\cos x

Another worked example

Differentiate h(x) = x times e to the x. Step 1: let the first factor be x with derivative 1, and the second be e to the x with derivative e to the x. Step 2: apply the rule to get (1) times e to the x plus x times e to the x. Step 3: simplify to h-prime(x) = e to the x times (1 plus x).

Aligned with the College Board AP Calculus CED — Unit 2 (Differentiation: Definition and Fundamental Properties), Topic 2.8 (The Product Rule). College Board ↗

Interactive Preview

Interactive simulator is loading...

Engaged with the simulation?

Practice actual exam problems and master calculus guaranteed. Get a 5 on the AP Calculus exam or get an A in the calculus class, or we'll pay you $100.

Accept the Mission