CalcRef

Derivatives · Basic Rules

Product Rule

ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

The derivative of a product: derivative of the first times the second, plus the first times the derivative of the second.

Worked examples

Find d/dx[x² sin x].
  1. f = x², f' = 2x, g = sin x, g' = cos x
  2. f'g + fg' = 2x sin x + x² cos x

Answer: 2x sin x + x² cos x

Find d/dx[eˣ · x].
  1. f = eˣ, f' = eˣ, g = x, g' = 1
  2. f'g + fg' = eˣ · x + eˣ · 1 = eˣ(x + 1)

Answer: eˣ(x + 1)

Related formulas

Quotient Rule

Practice this and 135 more formulas in the CalcRef workspace — quizzes, reference tables, a 16-category unit converter, and an expression evaluator.