Quotient Rule for Derivatives

The product rule handles functions multiplied together. Its partner, the quotient rule, handles one function divided by another: , , . The formula looks busier than the product rule, with a minus sign and a squared denominator, but it follows a fixed pattern, and once the pattern sticks it’s just bookkeeping.

What Is the Quotient Rule?

If and are differentiable and , then

In words: the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the bottom squared. You’ll also see it written with and :

Unlike the product rule, the two terms are subtracted, so their order matters. Swap them and every answer comes out with the wrong sign.

The Low D-High Mnemonic

Generations of calculus students have remembered it with a rhyme. Call the numerator “high” and the denominator “low”, and read “d” as “the derivative of”:

Low d-high minus high d-low, over low squared.

“Low d-high” is the bottom times the derivative of the top, . “High d-low” is the top times the derivative of the bottom, . “Low squared” is . The rhyme puts the terms in the right order, which is the only part that’s easy to get wrong.

Why the Minus Sign? Where the Rule Comes From

The quotient rule falls straight out of the product rule. Call the quotient , so that . Differentiate both sides with the product rule:

Now solve for , and replace with :

The minus sign has a plain meaning too. A fraction gets smaller when its denominator gets bigger, so the rate at which the bottom grows works against the derivative. That’s the term. (The rule can also be proved directly from the limit definition, but this route is shorter and shows why it looks the way it does.)

The Division Rule for Derivatives

You may see this rule called the division rule for derivatives, especially outside the US. It’s the same rule with a different name: the derivative of one function divided by another is

so everything on this page applies whichever name your course uses.

How to Use the Quotient Rule, Step by Step

1. Name the top and the bottom .
2. Differentiate each to get and .
3. Assemble , with brackets around every factor that has more than one term.
4. Simplify the numerator by expanding and collecting terms. Leave the denominator as a square: it’s never negative, so the sign of the derivative comes from the top alone, which is exactly what you need for finding where a function rises or falls.

Example 1: A Polynomial Over a Polynomial

Differentiate .

With , , , :

Watch the second term: the minus sign applies to the whole of , which is why the brackets matter.

Example 2: Checking Against the Power Rule

Differentiate with the quotient rule.

The top is the constant 1, so :

That matches the power rule on . Notice what happens if the terms are swapped: you’d get , a positive slope for a curve that’s clearly falling. A quick sanity check like this catches order mistakes.

Example 3: Finding Turning Points

Differentiate and find where the tangent line is horizontal.

A fraction is zero only when its numerator is zero, so solve : . The graph peaks at and bottoms out at . Because the denominator is always positive, the sign of tells you the rest: the function rises between and and falls outside that interval.

Example 4: The Derivative of tan x

Find .

Write . Using and :

The Pythagorean identity does the simplifying. The derivatives of trig functions get a lesson of their own later in this chapter.

Example 5: e to the x Over x

Differentiate .

Factoring out shows the slope is zero only at , negative for and positive for .

Example 6: A Linear Over a Linear

Differentiate .

The terms cancel completely. That always happens for a linear function over a linear function, and since the result is positive everywhere, this function is increasing on each side of its vertical asymptote at .

Example 7: Using a Table of Values

Suppose , , and . Find .

Careful with : two negatives make .

When You Don’t Need the Quotient Rule

The quotient rule always works, but it isn’t always the fastest:

• A constant denominator. is just , with derivative .
• A constant numerator. , so the power rule gives in one step.
• A single-term denominator. (for ), with derivative .

Save the quotient rule for fractions with the variable in both the top and the bottom.

Common Mistakes With the Quotient Rule

• Swapping the order. gives the negative of the right answer. Always start with the derivative of the top. If in doubt, test on .
• Forgetting to square the denominator. It’s , not .
• Dropping brackets in the second term. The minus sign applies to all of .
• Differentiating top and bottom separately. is L’Hôpital’s rule for limits, not the derivative of a quotient.
• Expanding the denominator. It almost never helps, and it hides the fact that the denominator is never negative.

Practice Problems

Try each one before opening the answer. The practice set has 12 more with instant checking.

Problem 1. Differentiate .

Problem 2. Differentiate .

Problem 3. Differentiate .

Problem 4. If , , and , find .

Problem 5. Find the tangent line to at .

With the power, product and quotient rules you can differentiate any combination of powers built with sums, products and quotients. The last basic rule, the chain rule, handles functions nested inside other functions, like or .