The quotient rule is a differentiation rule used when one function is divided by another function. It is especially useful for rational functions, trigonometric ratios, and expressions where both the numerator and denominator change with x. Students often remember it as the low d high minus high d low, all over low squared rule.
Knowing when to use it helps you differentiate complex fractions accurately and efficiently.
The rule comes from combining the product rule with the chain rule applied to a reciprocal. If y = f(x)/g(x), the derivative depends on both the rate of change of the numerator and the rate of change of the denominator. The denominator is squared because changing the bottom function affects the whole fraction nonlinearly.
In some problems, rewriting the quotient as a product with a negative exponent may be easier, but the quotient rule gives a direct and reliable method.
Key Facts
- If y = f(x)/g(x), then y' = [g(x)f'(x) - f(x)g'(x)]/[g(x)]^2.
- Memory phrase: low d high minus high d low, over low squared.
- The quotient rule applies when both numerator and denominator are functions of x.
- The denominator g(x) must not equal 0 at the point where the derivative is being evaluated.
- For y = u/v, dy/dx = (v du/dx - u dv/dx)/v^2.
- Some quotients can be simplified before differentiating, which may reduce algebra errors.
Vocabulary
- Quotient rule
- A differentiation rule for finding the derivative of one function divided by another function.
- Numerator
- The top part of a fraction, which is the function being divided.
- Denominator
- The bottom part of a fraction, which is the function doing the dividing.
- Derivative
- A function that gives the instantaneous rate of change or slope of another function.
- Rational function
- A function that can be written as a ratio of two polynomials.
Common Mistakes to Avoid
- Reversing the subtraction order: writing f'g - fg' instead of gf' - fg' changes the sign of the derivative.
- Forgetting to square the denominator: the correct bottom is [g(x)]^2, not just g(x), because the denominator also changes with x.
- Applying the power rule separately to the top and bottom: d[f/g]/dx is not f'/g', so differentiating each part alone gives the wrong result.
- Not simplifying before or after differentiating: leaving common factors or unsimplified expressions can hide mistakes and make final answers harder to use.
Practice Questions
- 1 Find dy/dx for y = (3x^2 + 1)/(x - 4).
- 2 Find the slope of f(x) = (2x + 5)/(x^2 + 1) at x = 1.
- 3 Decide whether the quotient rule or rewriting is more efficient for differentiating y = (x^3 - 2x)/x, and explain your reasoning.