Product And Quotient Rule

Product And Quotient Rule

Quotient Rule: 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.

In the previous section, we noted that we had to be careful when differentiating products or quotients. It’s now time to look at products and quotients and see why. 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.

Product And Quotient Rule

Quotient Rule Derivative

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.

Examples

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.

Derivative Quotient Rule

Derivative Quotient Rule

The following problems require the use of the quotient rule. In the following discussion and solutions, the derivative of a function h(x) will be denoted by tex2html_wrap_inline59 or h‘(x). The quotient rule is a formal rule for differentiating problems where one function is divided by another. It follows from the limit definition of derivative and is given by

tex2html_wrap_inline63 .

Remember the rule in the following way. Always start with the “bottom” function and end with the “bottom” function squared. Note that the numerator of the quotient rule is identical to the ordinary product rule except that subtraction replaces addition. In the list of problems that follows, most problems are average and a few are somewhat challenging.

PROBLEM 1: Differentiate tex2html_wrap_inline65 .

PROBLEM 2: Differentiate tex2html_wrap_inline67 .

PROBLEM 3: Differentiate tex2html_wrap_inline69 .

PROBLEM 4: Differentiate tex2html_wrap_inline71 .

PROBLEM 5: Differentiate tex2html_wrap_inline73 .

Quotient Rule Formula

If u and v are two functions of x, then the derivative of the product uv is given by…

Don’t miss…

Later in this section:
Quotient Rule

\displaystyle\frac{{{d}{\left({u}{v}\right)}}}

{{{\left.{d}{x}\right.}}}={u}\frac{{{d}{v}}}

{{{\left.{d}{x}\right.}}}+{v}\frac{{{d}{u}}}

{{\left.{d}{x}\right.}}

In words, this can be remembered as:

“The derivative of a product of two functions is the first times the derivative of the second, plus the second times the derivative of the first.”

Where does this formula come from? Like all the differentiation formulas we meet, it is based on derivatives from first principles.

Quotient Rule Calculus

Proof from derivative definition and limit properties

Let {\displaystyle f(x)=g(x)/h(x).} Applying the definition of the derivative and properties of limits gives the following proof.

Calculus Quotient Rule

First, let’s take a look at why we have to be careful with products and quotients. Suppose that we have the two functions f(x)=x3 and g(x)=x6. Let’s start by computing the derivative of the product of these two functions. This is easy enough to do directly.

(fg)′=(x3x6)′=(x9)′=9×8

Remember that on occasion we will drop the (x) part on the functions to simplify notation somewhat. We’ve done that in the work above.

Now, let’s try the following.

f′(x)

g′(x)=(3×2)(6×5)=18×7

So, we can very quickly see that.

(fg)′≠f′g′

In other words, the derivative of a product is not the product of the derivatives.

Using the same functions we can do the same thing for quotients.

(fg)′=(x3x6)′=(1×3)′=(x−3)′=−3

x−4=−3×4

f′

(x)g′(x)=3x26x5=12×3

So, again we can see that,

(fg)′≠f′g′

To differentiate products and quotients we have the Product Rule and the Quotient Rule.

What is the quotient rule in math?

The quotient rule is a formal rule for differentiating problems where one function is divided by another. It follows from the limit definition of derivative and is given by… Remember the rule in the following way. Always start with the “bottom” function and end with the “bottom” function squared.

How do you prove the quotient rule?

If you want to prove that 1g(x) is differentiable. Let r(x)=1g(x) and then use the definition of the derivative to show r′(x) exists. Then recall that you subtract fractions by finding a common denominator, which in this case is g2, and, you’ve got the quotient rule.

Why does the quotient rule work?

The quotient rule is a rule used to find the derivative of a function that can be written as the quotient of two functions. More simply, you can think of the quotient rule as applying to functions that are written out as fractions, where the numerator and the denominator are both themselves functions.