B.2 The Quotient Rule

On this screen we're going to develop "The Quotient Rule," which we need to find the derivative of the quotient of two differentiable functions, 𝑓𝑔. This is a natural next step after we found the Product Rule on the preceding screen. With this new rule, we'll be able to find the derivative of functions like 𝑒𝑥𝑥 and sin⁡𝜃cos⁡𝜃. We of course have Practice Problems below for you to use to become comfortable with this new rule.

As we saw for the Product Rule, let's emphasize immediately that the naive approach does not work: the derivative of the quotient of two functions is not the quotient of the derivatives:

After working to develop the Product Rule, we expect a more involved calculation than that of the naive approach. After all, we know from algebra that we can think of division as multiplication by the reciprocal of what's in the denominator: [𝑓(𝑥)𝑔(𝑥)]′=[𝑓(𝑥)⋅1𝑔(𝑥)]′ Hence we could think about the quotient instead as a product ... but to use this approach we need one more tool, which we'll see on the next screen.

Deriving the Quotient Rule

For now, let's stick with viewing the function 𝑓𝑔 as a quotient, and develop a Quotient Rule we can use to easily compute the derivative in terms of 𝑓, 𝑔, 𝑓′ and 𝑔′. We begin, of course, with the Definition of the Derivative, now applied to the quotient 𝑓𝑔 :

Let's focus briefly on what's in the dashed box, setting aside the limit and 1ℎ for a moment. Similar to what we encountered in our development of the Product Rule, it's unclear how to proceed given this collection of terms, especially those in the numerator. But let's remember that we're aiming to develop a Quotient Rule that involves the terms 𝑓′(𝑥) and 𝑔′(𝑥), and so we can work to force those to appear. In fact, in our "algebra-based derivation of the Product Rule," we did just that, using an "algebraic trick" to cleverly introduce the terms we need: we subtract and add the same term 𝑓(𝑥) 𝑔(𝑥) to force the definitions of the derivatives 𝑓′(𝑥) and 𝑔′(𝑥) to pop up. Let's do the same thing here:

The bits in blue and green, the first and last [...] terms in the numerator, will turn into 𝑓′(𝑥) and 𝑔′(𝑥) respectively when we divide by h and take the limit in a minute. Substituting back into the equation (*) where we left off above,

As you saw, in the second-to-last line when we evaluated the limit, the pesky h in the denominator went away as part of the definition of the derivative for 𝑓′ and 𝑔′, because of the algebraic trick we used. We have thus developed the Quotient Rule we were after:

This rule is usually harder for students to remember than, say, the Product Rule, so be on guard. Typical mistakes include switching the terms in the numerator, and forgetting to square the denominator.

Many students remember the quotient rule by thinking of the numerator as "high," the demoninator as "low," the derivative as "d," and then singing:

♫ "low d-high,
minus high d-low,
all over low-squared" ♫

♫ "low d-high, minus high d-low,
all over low-squared" ♫

Every teacher who's given an exam on this material has seen students silently mouthing that phrase — to good use! — to make sure they use the Quotient Rule correctly.

The Quotient Rule may seem more intimidating to use than the Product Rule was, but we promise that with some practice (see below!) you'll quickly get it down and have it as part of your working toolkit. Let's consider a few examples to see how it works.

The result of Example 2 is actually a big "trig function derivative" that we'll use on occasion, and that appears in our "Trig Function Derivatives" and "Table of Derivatives" screens that are accessible from the "Key Formulas" menu at the top of every screen:

As we wrote above, while the Quotient Rule may initially look awkward to use, with practice it will feel as routine as using something like the Quadratic Formula has probably become for you. So let's practice!

Practice Problems