Calculus:
Derivative: Quotient Rule

1.  
\(\frac{d}{dx} \frac{x}{\cos{x}}\)
  Solution
2.  
\(\frac{d}{dx} \frac{\ln{x}}{x}\)
  Solution
3.  
\(\frac{d}{dx} \frac{{x}^{2}+3}{\tan{x}}\)
  Solution
4.  
\(\frac{d}{dx} \frac{{x}^{2}}{3x-1}\)
  Solution
5.  
\(\frac{d}{dx} \frac{4\sin{x}}{x+\cos{x}}\)
  Solution
6.  
\(\frac{d}{dx} \frac{{2}^{x}}{{2}^{x}-{3}^{x}}\)
  Solution
7.  
\(\frac{d}{dx} \frac{\cos{x}}{{x}^{2}}\)
  Solution
8.  
\(\frac{d}{dx} \frac{{x}^{2}+1}{{e}^{x}}\)
  Solution

Quotient Rule for Derivatives - Introduction

If you are looking for the derivative of a function, sometimes you might not know where to start. Fortunately, for most functions, there are a set of rules that you can apply to lead to the solution. We will now discuss the case where the expression is a fraction, with one sub-expression in the numerator and another in the denominator.
In this case, the rule to apply is the quotient rule for derivatives. You can use it when your function takes the form
\(\frac{f}{g}\)
. Note that after applying the quotient rule, you will not be done yet. Instead, the expression will have been broken down into parts, where the derivatives of these parts will be easier to find. You can then move on to apply other rules, such as the power rule or the constant rule, to these smaller parts.
Let’s look at an example in the next section.

Using the Quotient Rule for Derivatives

Consider the function
\(\frac{\ln{x}}{x}\)
. The quotient rule states that:
\((\frac{f}{g})'=\frac{f'g-fg'}{{g}^{2}}\)
where the apostrophe (
\('\)
) means "the derivative of".
In this example, we let
\(f=\ln{x}\)
and
\(g=x\)
.
This allows us to rewrite the function as:
\(\frac{((\frac{d}{dx})\ln{x})x-\ln{x}((\frac{d}{dx})x)}{{x}^{2}}\)
Other calculus rules tell us that the derivative of
\(\ln{x}\)
is always
\(\frac{1}{x}\)
. As a result, the first half of our numerator becomes
\(x(\frac{1}{x})\)
, which neatly simplifies to
\(1\)
. This gives:
\(\frac{1-\ln{x}((\frac{d}{dx})x)}{{x}^{2}}\)
Now we can use the power rule, which states that
\((\frac{d}{dx}){x}^{n}=n{x}^{n-1}\)
. Since we know that
\(n\)
in this case is
\(1\)
(because
\(x\)
has no exponent), this becomes
\(1\times {x}^{1-1}\)
, which yields a value of
\(1\)
. This gives us our final answer:
\(\frac{1-\ln{x}}{{x}^{2}}\)

Quotient Rule - Calculus Practice Problems

Still not sure about the quotient rule? Try some of our practice problems at the top of this page, and use the step-by-step solutions if you get stuck. When it comes to the quotient rule in calculus, don’t be surprised if you need to leverage several other rules to find the final derivative. The sum rule, power rule and the chain rule are all commonly used when you’re solving for the derivative of a function. See our other practice problems to find out more.

What's Next

At Cymath, we believe that working through examples is one of the best ways to gain mastery of the quotient rule for derivatives. We hope that when those examples are accompanied by clear explanations and step-by-step instructions, they can help your understanding. You can also subscribe to Cymath Plus, which offers ad-free and more in-depth help, from pre-algebra to calculus.
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