The Product Rule: Interactive Slope Builder

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'.

Formula

\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}

Easy Example Problem

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

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