Quotient Rule: Formula & Examples

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  • 0:03 Definition and Formula
  • 0:53 Mnemonic Device
  • 1:40 Examples
  • 4:02 Lesson Summary
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Lesson Transcript
Miriam Snare

Miriam has taught middle- and high-school math for over 10 years and has a master's degree in Curriculum and Instruction.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

In this lesson, you will learn the formula for the quotient rule of derivatives. The lesson includes a mnemonic device to help you remember the formula. You will also see two worked-out examples.

Definition and Formula

The quotient rule is a formula for taking the derivative of a quotient of two functions. It makes it somewhat easier to keep track of all of the terms. Let's look at the formula.

If you have function f(x) in the numerator and the function g(x) in the denominator, then the derivative is found using this formula:

Quotient Rule Formula

In this formula, the d denotes a derivative. So, df(x) means the derivative of function f and dg(x) means the derivative of function g. The formula states that to find the derivative of f(x) divided by g(x), you must:

  1. Take g(x) times the derivative of f(x).
  2. Then from that product, you must subtract the product of f(x) times the derivative of g(x).
  3. Finally, you divide those terms by g(x) squared.

Mnemonic Device

The quotient rule formula may be a little difficult to remember. Perhaps a little yodeling-type chant can help you. Imagine a frog yodeling, 'LO dHI less HI dLO over LO LO.' In this mnemonic device, LO refers to the denominator function and HI refers to the numerator function.

Let's translate the frog's yodel back into the formula for the quotient rule.

LO dHI means denominator times the derivative of the numerator: g(x) times df(x).

less means 'minus'.

HI dLO means numerator times the derivative of the denominator: f(x) times dg(x).

over means 'divide by'.

LO LO means take the denominator times itself: g(x) squared.


Let's look at a couple of examples where we have to apply the quotient rule.

In the first example, let's take the derivative of the following quotient:


Let's define the functions for the quotient rule formula and the mnemonic device. The f(x) function (the HI) is x^3 - x+ 7. The g(x) function (the LO) is x^2 - 3.

Now, let's take the derivative of each function. df(x), or dHI, is 3x^2 - 1. dg(x), or dLO, is 2x.

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Additional Activities

Practice Problems - Quotient Rule

In the following practice problems, students will use the quotient rule to find the derivatives of various functions. Students will also use the quotient rule to show why the derivative of tangent is secant squared.


Use the quotient rule to differentiate the following functions. Simplify number 1 as much as possible. Do not simplify number 2.

3. Use the quotient rule to prove that


1. Using the quotient rule, we have

Then, distribute in the numerator and combine like terms to simplify.

2. Using the quotient rule, and remembering that the derivative of sine is cosine, we have

3. To show that the derivative of tangent is secant squared, first rewrite tangent in terms of sine and cosine.

Now use the quotient rule. Remember that the derivative of cosine is negative sine.
Now simplify using the trigonometric identity
to get
Then, since secant is equal to one over cosine, we have
and so the formula for the derivative of tangent is proven.

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