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The Quotient Rule

Abigail Young

Abigail Young

5 min read

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Study Guide Overview

This study guide covers the Quotient Rule for finding derivatives of rational functions. It provides the definition and a memory aid (Low D-High, High D-Low...). Several examples demonstrate the application and simplification of the Quotient Rule, including trigonometric and exponential functions. Practice problems and solutions reinforce understanding. Key takeaways and a formula image are included.

The Quotient Rule 🚀

Hey there, future calculus master! Ready to tackle the Quotient Rule? This is a key tool for differentiating those tricky rational functions, and you'll see it pop up all over the AP exam. Let's make sure you're totally comfortable with it!


⛳ Quotient Rule Definition

The Quotient Rule helps us find the derivative of a function that's a ratio of two other functions. Here's the formal definition:

ddx[f(x)g(x)]=g(x)ddxf(x)−f(x)ddxg(x)(g(x))2\frac{d}{dx} \Bigg[\frac{f(x)}{g(x)}\Bigg] = \frac{g(x)\frac{d}{dx}f(x)-f(x)\frac{d}{dx}g(x)}{(g(x))^2}


Memory Aid

Low D-High, High D-Low, Draw the Line and Square Below! 🎶 Think of it like a little song.

  • Low: The denominator function, g(x)
  • D-High: Derivative of the numerator function, f'(x)
  • High: The numerator function, f(x)
  • D-Low: Derivative of the denominator function, g'(x)
  • Square Below: The denominator squared, (g(x))^2

Or, if you prefer a more compact version using u and v:...

Previous Topic - The Product RuleNext Topic - Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions

Question 1 of 9

What is the correct formula for the Quotient Rule when finding the derivative of f(x)g(x)\frac{f(x)}{g(x)}? 🤔

f′(x)g(x)−f(x)g′(x)(g(x))2\frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}

f(x)g′(x)−g(x)f′(x)(g(x))2\frac{f(x)g'(x) - g(x)f'(x)}{(g(x))^2}

g(x)f′(x)+f(x)g′(x)(g(x))2\frac{g(x)f'(x) + f(x)g'(x)}{(g(x))^2}

g(x)f′(x)−f(x)g′(x)(g(x))2\frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}