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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.

  • 2 worked examples
  • 24 in Derivatives

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

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Find d/dx[eˣ · x].

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Quotient Rule

See all 24 Derivatives formulas on one printable sheet - the same statements and conditions as these pages, typeset and laid out for paper.

Common questions

What is the Product Rule?

The derivative of a product: derivative of the first times the second, plus the first times the derivative of the second. It is one of the derivatives formulas in the CalcRef reference.

How do you use the Product Rule?

Worked example. Find d/dx[x² sin x]. f = x², f' = 2x, g = sin x, g' = cos x. f'g + fg' = 2x sin x + x² cos x. Answer: 2x sin x + x² cos x.

Practice

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