Quotient Rule
Derivative of a quotient of two functions.
Formula
Variables
Example
Differentiates fractions of functions
How it works
The quotient rule tells you how to differentiate a ratio of two changing functions, like one curve divided by another. It comes from rewriting the quotient as a product, u \cdot v^{-1}, and then applying the product rule plus the chain rule; that is why both u' and v' appear and why the denominator is squared. The numerator combines the two rates of change in a specific order so the result measures how the fraction changes as x changes.
When to use it
Use this when a function is explicitly written as one function divided by another and it is awkward or less convenient to simplify first. It appears often in physics, economics, and calculus problems involving rates, densities, averages, or rational functions.
Worked example, step by step
Differentiate f(x) = (x^2 + 1)/(x - 3) Let u = x^2 + 1, so u' = 2x Let v = x - 3, so v' = 1 Apply the quotient rule: f'(x) = (u'v - uv') / v^2 f'(x) = [2x(x - 3) - (x^2 + 1)(1)] / (x - 3)^2 Expand the numerator: 2x^2 - 6x - x^2 - 1 Simplify: f'(x) = (x^2 - 6x - 1) / (x - 3)^2
Common mistakes to avoid
- Reversing the order in the numerator and writing uv' - u'v instead of u'v - uv'
- Forgetting to square the entire denominator
- Differentiating the numerator and denominator separately without using the quotient rule
Did You Know?
The quotient rule can always be avoided by rewriting a fraction as a product with a negative exponent.
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IntermediateDerivative of a power function.
Chain Rule
IntermediateDerivative of a composite function.
Product Rule
IntermediateDerivative of a product of two functions.
Quotient Rule
IntermediateDerivative of a quotient of two functions.
Derivative of Sine
IntermediateDerivative of the sine function.
Derivative of Cosine
IntermediateDerivative of the cosine function.