The Quotient Rule - AP Calculus BC Study Guide

The Core Idea: The Quotient Rule

The quotient rule addresses the challenge of finding the derivative of a function that is constructed by dividing two other differentiable functions. Just as with products of functions, the derivative of a quotient is not simply the derivative of the numerator divided by the derivative of the denominator. Instead, a specific formula is required to correctly determine the instantaneous rate of change of the overall function.

This rule provides a precise method for calculating this derivative by combining the original functions and their respective derivatives in a structured way. It is a fundamental tool for differentiating a wide class of functions, particularly rational functions and combinations of trigonometric, exponential, and polynomial functions. Mastery of the quotient rule is essential for finding derivatives of more complex expressions.

Key Formulas

The derivative of a quotient of two differentiable functions is found using the quotient rule. If a function $h(x)$ is defined as the quotient of two functions $f(x)$ and $g(x)$, such that $h(x)=\frac{f(x)}{g(x)}$, its derivative is given by:

$$h^{\prime }(x)=\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$$

To aid in memorization, this formula can be remembered as: "Low d-High minus High d-Low, all over the square of what's below," where "Low" refers to the denominator $g(x)$, "High" refers to the numerator $f(x)$, and "d" signifies taking the derivative.

Understanding the Application

The primary condition for applying the quotient rule is that the function must be a quotient of two differentiable functions. The rule is specifically designed for functions of the form $h(x)=\frac{\text{numerator function}}{\text{denominator function}}$.

A critical nuance of the quotient rule is the subtraction in the numerator: $g(x)f^{\prime }(x)-f(x)g^{\prime }(x)$. Unlike the product rule, where the terms are added, the order of these terms is crucial. Reversing the order of subtraction will result in the negative of the correct answer. Therefore, it is essential to always begin the numerator with the denominator function ($g(x)$) multiplied by the derivative of the numerator function ($f^{\prime }(x)$). The denominator of the derivative is always the square of the original denominator function, $[g(x){]}^2$.

Core Concepts & Rules

• Purpose: The quotient rule is used to find the derivative of a function that is the ratio of two differentiable functions.

• Structure: The rule applies to functions of the form $h(x)=\frac{f(x)}{g(x)}$.

• Required Components: To apply the rule, you need four pieces of information: the numerator function ($f(x)$), the denominator function ($g(x)$), the derivative of the numerator ($f^{\prime }(x)$), and the derivative of the denominator ($g^{\prime }(x)$).

• The Formula: The derivative is calculated as $h^{\prime }(x)=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$.

• Order Matters: The subtraction in the numerator makes the order of operations critical. The term $g(x)f^{\prime }(x)$ must come first.

Step-by-Step Example 1: Basic Application

Problem: Find the derivative of the function $h(x)=\frac{5x^2}{e^x}$.

Step 1: Identify the numerator and denominator functions.

Let the numerator be $f(x)=5x^2$.

Let the denominator be $g(x)=e^x$.

Step 2: Find the derivatives of the numerator and denominator.

Using the power rule, the derivative of the numerator is $f^{\prime }(x)=10x$.

The derivative of the denominator is $g^{\prime }(x)=e^x$.

Step 3: Substitute these four components into the quotient rule formula.

The formula is $h^{\prime }(x)=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$.

Substituting our functions and their derivatives:

$$h^{\prime }(x)=\frac{(e^x)(10x)-(5x^2)(e^x)}{[e^x{]}^2}$$

Step 4: Simplify the resulting expression.

Factor out the common term $e^x$ from the numerator.

$$h^{\prime }(x)=\frac{e^x(10x-5x^2)}{(e^x)(e^x)}$$

Cancel one $e^x$ term from the numerator and denominator.

$$h^{\prime }(x)=\frac{10x-5x^2}{e^x}$$

Step-by-Step Example 2: Exam-Style Application

Problem: Let $k(x)=\frac{f(x)}{g(x)}$. The functions $f$ and $g$ are differentiable. The table below gives selected values of the functions and their derivatives. Find the value of $k^{\prime }(1)$.

Step 1: Write down the general formula for the derivative of $k(x)$.

Using the quotient rule:

$$k^{\prime }(x)=\frac{g(x)f^{\prime }(x)-f(x)g^{\prime }(x)}{[g(x){]}^2}$$

Step 2: Evaluate the formula at the specific point, $x=1$.

$$k^{\prime }(1)=\frac{g(1)f^{\prime }(1)-f(1)g^{\prime }(1)}{[g(1){]}^2}$$

Step 3: Extract the necessary values from the provided table.

From the row where $x=1$:

• $f(1)=-3$

• $f^{\prime }(1)=4$

• $g(1)=2$

• $g^{\prime }(1)=5$

Step 4: Substitute the numerical values into the formula and compute the result.

$$k^{\prime }(1)=\frac{(2)(4)-(-3)(5)}{[2{]}^2}$$

$$k^{\prime }(1)=\frac{8-(-15)}{4}$$

$$k^{\prime }(1)=\frac{8+15}{4}$$

$$k^{\prime }(1)=\frac{23}{4}$$

Using Your Calculator

The quotient rule is an analytical technique that must be performed by hand, especially on the no-calculator portion of the AP exam. A graphing calculator cannot perform the symbolic differentiation required to show the steps of the quotient rule.

However, on the calculator-active portion of the exam, you can use the calculator's numerical derivative feature to check your answer or to find the value of a derivative at a specific point without showing the analytical steps.

To verify the result of Example 2:

To find $k^{\prime }(1)$ for $k(x)=\frac{f(x)}{g(x)}$, if you were given explicit functions (e.g., $f(x)=4x-7$ and $g(x)=5x-3$), you could:

1. Enter the function $k(x)=\frac{4x-7}{5x-3}$ into $Y_1$.

2. Use the numerical derivative command from the calculator's MATH menu. The syntax is typically $nDeriv(expression,variable,value)$ or a template that looks like $\frac{d}{dx}(\square ){|}_{x=\square }$.

3. On the home screen, you would input nDeriv(Y_1, X, 1).

4. The calculator would return a numerical approximation of the derivative's value at x=1, which you can compare to your by-hand calculation.

AP Exam Quick Hit

Common Question Types

• Direct Differentiation: You will be asked to find the derivative of a function presented as a quotient, such as finding $y^{\prime }$ for $y=\frac{\ln (x)}{x^3+2x}$. This tests direct application of the formula and simplification.

• Finding the Equation of a Tangent Line: You might be asked to find the equation of the line tangent to the graph of $h(x)=\frac{f(x)}{g(x)}$ at $x=c$. This requires using the quotient rule to find the slope, $h^{\prime }(c)$, and evaluating $h(c)$ to find the point of tangency.

• Using Tabular or Graphical Data: As in Example 2, you will be given a table of values or graphs of two functions, $f$ and $g$, and asked to find the derivative of their quotient at a specific point. This assesses your ability to apply the rule's structure without complex algebra.

Common Mistakes

• Incorrect Order in Numerator: The most common error is reversing the terms in the numerator, calculating $f(x)g^{\prime }(x)-g(x)f^{\prime }(x)$. This results in the negative of the correct answer. Always start with "Low d-High."

• Forgetting to Square the Denominator: Students sometimes forget to square the denominator, leaving just $g(x)$ instead of $[g(x){]}^2$.

• The "Fake" Quotient Rule: A major conceptual error is to assume the derivative of a quotient is the quotient of the derivatives: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)\ne \frac{f^{\prime }(x)}{g^{\prime }(x)}$.

• Algebraic Simplification Errors: After correctly applying the rule, students often make mistakes when simplifying the resulting expression, particularly with distributing negative signs in the numerator or handling complex fractions.

• Product Rule vs. Quotient Rule: Confusing the product rule formula with the quotient rule formula, such as using a $+$ sign in the numerator instead of a $-$ sign.