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2.9 The Quotient Rule

Part of Differentiation: Definition and Basic Derivative Rules in AP® Calculus BC. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The quotient rule is used when one function is divided by another. If
y=f(x)g(x)y=\frac{f(x)}{g(x)}
,
then its derivative is
y′=f′(x)g(x)−f(x)g′(x)[g(x)]2y'=\frac{f'(x)g(x)-f(x)g'(x)}{[g(x)]^2}
,
provided
g(x)≠0g(x)\ne0
.
The numerator follows the pattern “low d-high minus high d-low” if you name the denominator the low function and the numerator the high function. The real goal is understanding the structure rather than memorizing a phrase mechanically. The denominator is squared because the derivative of a quotient depends on how changes in the denominator affect the whole fraction.

In this unit

10 lessons

  1. 2.1Defining Average and Instantaneous Rates of Change at a PointPreview lesson
  2. 2.2Defining the Derivative of a Function and Using Derivative NotationPreview lesson
  3. 2.3Estimating Derivatives of a Function at a PointPreview lesson
  4. 2.4Connecting Differentiability and Continuity: Determining When Derivatives Do and Do Not ExistPreview lesson
  5. 2.5Applying the Power RulePreview lesson
  6. 2.6Derivative Rules: Constant, Sum, Difference, and Constant MultiplePreview lesson
  7. 2.7Derivatives of cos x, sin x, eˣ, and ln xPreview lesson
  8. 2.8The Product RulePreview lesson
  9. 2.9The Quotient RuleThis lesson
  10. 2.10Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant FunctionsPreview lesson

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