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Activity5.8The Quotient Rule

The next rule we are going to explore is called the quotient rule of differentiation. We never use this rule unless there is a variable factor in the denominator of the expression we are differentiating. (Remember, we already have the constant factor/divisor rule to deal with constant factors in the denominator.)\begin{equation}\lzoo{x}{\frac{\fe{f}{x}}{\fe{g}{x}}}=\frac{\lzoo{x}{\fe{f}{x}}\cdot\fe{g}{x}-\fe{f}{x}\cdot\lzoo{x}{\fe{g}{x}}}{\left[\fe{g}{x}\right]^2}\label{equation-quotient-rule}\tag{5.8.1}\end{equation}

Like all of the other rules, you ultimately want to perform the quotient rule in your head. Your instructor, however, may initially want you to show steps when applying the quotient rule; under that presumption, steps are going to be shown in each and every example of this lab when the quotient rule is applied. Two simple examples of the quotient rule are shown in Table 5.8.1.

Given function Derivative
\(y=\dfrac{4x^3}{\fe{\ln}{x}}\phantom{\frac{\lzoo{x}{4x^3}}{\left[\fe{\ln}{x}\right]^2}}\) \(\begin{aligned}[t]\lz{y}{x}&=\frac{\lzoo{x}{4x^3}\cdot\fe{\ln}{x}-4x^3\cdot\lzoo{x}{\fe{\ln}{x}}}{\left[\fe{\ln}{x}\right]^2}\\&=\frac{12x^2\cdot\fe{\ln}{x}-4x^3\cdot\frac{1}{x}}{\left[\fe{\ln}{x}\right]^2}\\&=\frac{12x^2\fe{\ln}{x}-4x^2}{\left[\fe{\ln}{x}\right]^2}\end{aligned}\)
\(V=\dfrac{\fe{\csc}{t}}{\fe{\tan}{t}}\phantom{\frac{\lzoo{t}{\fe{\csc}{t}}}{\fe{\tan^2}{t}}}\) \(\begin{aligned}[t]\lz{V}{t}&=\frac{\lzoo{t}{\fe{\csc}{t}}\cdot\fe{\tan}{t}-\fe{\csc}{t}\cdot\lzoo{t}{\fe{\tan}{t}}}{\fe{\tan^2}{t}}\\&=\frac{-\fe{\csc}{t}\fe{\cot}{t}\fe{\tan}{t}-\fe{\csc}{t}\fe{\sec^2}{t}}{\fe{\tan^2}{t}}\\&=\frac{-\fe{\csc}{t}\left(1+\fe{\sec^2}{t}\right)}{\fe{\tan^2}{t}}\end{aligned}\)
Table5.8.1Examples of the Quotient Rule

Subsection5.8.1Exercises

Find the first derivative formula for each of the following functions. In each case take the derivative with respect to the independent variable as implied by the expression on the right side of the equal sign. Make sure that you use the appropriate name for each derivative.

1

\(\fe{g}{x}=\dfrac{4\fe{\ln}{x}}{x}\)

2

\(\fe{j}{y}=\dfrac{\sqrt[3]{y^5}}{\fe{\cos}{y}}\)

3

\(y=\dfrac{\fe{\sin}{x}}{4\fe{\sec}{x}}\)

4

\(\fe{f}{t}=\dfrac{t^2}{e^t}\)

Find each of the following derivatives without first simplifying the formula; that is, go ahead and use the quotient rule on the expression as written. For each derivative, check your answer by simplifying the original expression and then taking the derivative of that simplified expression.

5

\(\displaystyle\lzoo{t}{\dfrac{\fe{\sin}{t}}{\fe{\sin}{t}}}\)

6

\(\displaystyle\lzoo{x}{\dfrac{x^6}{x^2}}\)

7

\(\displaystyle\lzoo{x}{\dfrac{10}{2x}}\)