Just like a product is the result of a multiplication, a “quotient” is a result of a division. Therefore, the Quotient Rule is used to calculate the derivative of the division of functions.

The Quotient Rule is used when two functions are dividing. You know two functions are dividing when they look like a fraction, with one function on top and one on bottom, with a horizontal line between them. Here are some examples of dividing functions:

$$f(x)=\frac{x^3+4x}{2x^2}$$

$$f(x)=\frac{\sqrt{x}}{x+5}$$

$$f(x)=\frac{3^x+9}{x+3}$$

See how they all look like fractions? That means that the function on the bottom is diving the function on top. In business calculus, you probably will never have to learn how to divide functions per se. Instead, what you need to know is how to get the derivative of two dividing functions, for which you use the Quotient Rule.

The Quotient Rule is very similar to the Product Rule, with the exception that you subtract instead of add, and that you divide the top result by the bottom function squared at the end.

In words: The derivative of a quotient (or division) of functions is equal to the bottom function times the derivative of the top function, minus the top function times the derivative of the bottom function, with everything divided over the bottom function squared

In math notation: If $q(x)=\frac{f(x)}{g(x)}$ then $q'(x)=\frac{g(x) \cdot f'(x) - f(x) \cdot g'(x)}{(g(x))^2}$

Here's the process broken down into concrete steps:

- Step 1: Multiply the bottom function times the derivative of the top function
- Step 2: Multiply the top function times the derivative of the bottom function
- Step 3: Subtract the result of Step 2 from the result of Step 1
- Step 4: Divide the result of Step 3 by the bottom function squared (elevated to the second power)

Below, we will explore a problem using the steps outlined above.

Given that $f(x)= \frac{3x^2+20x}{2x^3+5}$

Question 1Find $f'(x)$

You know you need to use the Quotient Rule with this function because it's a fraction, with a function on top and another function on the bottom.

Beginning with Step 1, we multiply the bottom function times the derivative of the top function,

$$(2x^3+5) \cdot (3x^2+20x)'$$

$$=(2x^3+5)(6x+20)$$

We simplify this term by using FOIL,

$$=12x^4+40x^3+30x+100$$

Then, we move on to Step 2, and multiply the top function times the derivative of the bottom function,

$$(3x^2+20x) \cdot (2x^3+5)'$$

$$=(3x^2+20x) \cdot (6x^2)$$

We also simplify this term by distributing the multiplication,

$$=18x^4+120x^3$$

Then, we'll combine Steps 3 and 4. For Step 3, we subtract the results of Step 2 from Step 1 (which becomes the top):

$$12x^4+40x^3+30x+100-$$

$$(18x^4+120x^3)$$

Here, we need to distribute the negative sign to the $18x^4+120x^3$, making both terms negative:

$$12x^4+40x^3+30x+100-$$

$$18x^4-120x^3$$

Then, we simplify by combining like terms,

$$=-6x^4-80x^3+30x+100$$

Finally, as per Step 3, we divide this by the bottom function squared, to get our final answer:

$$=\frac{-6x^4-80x^3+30x+100}{(2x^3+5)^2}$$

- The Quotient Rule is used to find the derivative of dividing functions
- The Quotient Rule says the derivative of a division of functions is equal to the bottom function times the derivative of the top function, minus the top function times the derivative of the bottom function, with everything divided by the bottom function squared

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