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The product rule finds the derivative of two functions multiplied together. If h(x) = f(x) · g(x), then h'(x) = f'(x) · g(x) + f(x) · g'(x) — "derivative of first times second, plus first times derivative of second."
A common mistake is to differentiate each factor separately and multiply: d/dx[f·g] ≠ f'·g'. The product rule is needed because the rate of change of a product depends on both factors changing simultaneously.
Worked example: differentiate h(x) = x² · sin(x). Let f = x² (so f' = 2x) and g = sin(x) (so g' = cos(x)). Then h'(x) = 2x · sin(x) + x² · cos(x).
The product rule extends to three or more factors: d/dx[f·g·h] = f'gh + fg'h + fgh'. It also combines with the chain rule for expressions like x³ · e^(2x).
d/dx[f(x)·g(x)] = f'(x)·g(x) + f(x)·g'(x)
Quick examples
Product Rule Formula
d/dx[f·g] = f'g + fg'
“Derivative of first times second, plus first times derivative of second.”
This calculator is for educational purposes only and does not constitute professional advice. Results are based on standard mathematical formulas. Always verify critical calculations with a qualified professional before making important decisions.