Calculus
Class 12

Product Rule

ddx[f(x)g(x)]=f(ˊx)g(x)+f(x)g(ˊx)\frac{d}{dx}[f(x)g(x)] = f\'(x)g(x) + f(x)g\'(x)
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d/dx[f×g] = f'g + fg'

Derivative of a product of two functions. "First times derivative of second plus second times derivative of first."

Real-World Applications

Physics — Momentum change: d/dt(mv) when both mass and velocity change.

Economics — Revenue optimization: R = P × Q, both functions of time.

Engineering — Power in circuits: P = V × I.

Biology — Growth rate when birth and death rates both change.

Finance — Portfolio value: sum of (shares × price).

Chemistry — Reaction rates with changing concentrations.

Marketing — Sales = (reach × conversion rate).

Statistics — Joint probability derivatives.

Manufacturing — Production rate × quality rate.

Environmental science — Pollution = source rate × dispersion.

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