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Math · Calculus I · Concept

The quotient rule for derivatives

The quotient rule gives the derivative of f/g: (f/g)′ = (f′g − fg′)/g², wherever g ≠ 0. The order in the numerator matters, because of the minus sign; simplify the numerator and keep the denominator squared.

The rule and its order

For differentiable f and g with g(x) ≠ 0, the derivative of f/g is the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared. Swapping the two terms of the numerator changes the sign of the answer.

(f⁡g⁡)′=f⁡′g⁡−f⁡g⁡′g⁡2

Where it comes from

Write f/g as f·g⁻¹. The product rule gives f′·g⁻¹ + f·(g⁻¹)′, and the chain rule gives (g⁻¹)′ = −g⁻²·g′. Putting both terms over g² gives the quotient rule.

(f⁡g⁡−1)′=f⁡′g⁡−1−f⁡g⁡−2g⁡′=f⁡′g⁡−f⁡g⁡′g⁡2
(f⁡g⁡−1)′=f⁡′g⁡−1−f⁡g⁡−2g⁡′=f⁡′g⁡−f⁡g⁡′g⁡2

Simplify the numerator, keep the denominator

Expand and combine the numerator, and leave the denominator as g² in factored form unless it cancels with a factor of the numerator. The derivative has the same excluded values as the original function.

When you do not need the quotient rule

A constant denominator is just a constant multiple: (x² + 1)/5 has derivative 2x/5. A single power in the denominator can be rewritten: 3/x² = 3x⁻² has derivative −6x⁻³. Use the quotient rule when the denominator is a genuine function of x.

Common mistakes

  • Reversing the numerator: fg′ − f′g gives the negative of the correct derivative.
  • Forgetting to square the denominator.
  • Differentiating the top and bottom separately: (f/g)′ is not f′/g′.
  • Dropping the excluded values: the derivative is undefined wherever g(x) = 0.

Key terms

Quotient rule
(u/v)′ = (u′v − uv′)/v² where v ≠ 0: the bottom times the derivative of the top, minus the top times the derivative of the bottom, over the bottom squared. Values that made the original denominator zero stay excluded.
Product rule
To differentiate a product, take the derivative of the first factor times the second, plus the first times the derivative of the second: (uv)′ = u′v + uv′.
Chain rule
The rule for differentiating a function inside another function: differentiate the outside, keeping the inside as it is, then multiply by the derivative of the inside.
Rational expression
A fraction whose numerator and denominator are polynomials. Any value that makes the original denominator zero stays excluded, even if that factor cancels when you simplify.
Domain
The set of inputs for which a function or expression is defined. In the real numbers that rules out zero denominators, negative numbers under even roots and inputs of logarithms that aren’t positive.

Work through an example

Differentiate f(x) = (x² + 1)/(x − 3), then find where its tangent line is horizontal.

Differentiate a quotient of polynomials →
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Make it concrete

Try in the workspace

Open the example inputs, change a value and keep a useful result on your board.

Check the derivative in Math See the horizontal tangents in Graph Open worked example on a board Quotient rule in Math Reference

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