Introduction

Welcome everyone once again to another day of learning Calculus! I am assuming you just read the title and realized that this is basically the product rule, but its division. 

The Quotient Rule 

$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right]=\frac{f’(x)g(x)-f(x)g’(x)}{(g(x){)}^2}$

Same with the product rule, you can simply call the functions $f$ and $g$, like this:

$\frac{d}{dx}\left[\frac{f}{g}\right]=\frac{f’g-g’f}{g^2}$

Notice the similarities between the product and the quotient rule. There are a few differences, like the minus sign in the numerator, and the noticeable $g(x{)}^2$ in the denominator. There are also few similarities, like the product rule being in the numerator. 

This is an extremely similar rule to the product rule that doesn’t need an exceptional amount of elaborating compared to previous articles. We will do one example, and the rest is yours!

Example 1:

$f(x)=\frac{x}{x^2+1}$. Solve for $f’(2)$.

One thing that you will still have to do is to identify the individual functions, and solve for their derivatives. The distinction between the individual functions is very clear, just take the numerator as $f(x)$ and the denominator as $g(x)$. List out each function in a nice and orderly manner, and simply plug things into the formula!

Something that I would like to note is that you cannot switch around $f(x)$ and $g(x)$ with the quotient rule. $f(x)$ is STRICTLY $x$ because it is in the numerator, and $g(x)$ is STRICTLY $x^2+1$ because it is in the denominator. Now that we understand this, we can start solving. 

$f(x)=x$

$g(x)=x^2+1$

$f’(x)=1$

$g’(x)=2x$

Plug the functions into the quotient rule. This is all stuff you’ve done with the product rule, just with a different rule.

$\frac{f’g-g’f}{g^2}=\frac{(1)(x^2+1)-(2x)(x)}{(x^2+1{)}^2}=\frac{x^2+1-2x^2}{(x^2+1{)}^2}=\frac{1-x^2}{(x^2+1{)}^2}$

We have $f’(x)$, and now we can solve for $f’(2)$.

$f(2)=\frac{1-(2{)}^2}{((2{)}^2+1{)}^2}=\frac{1-4}{(5{)}^2}=\frac{-3}{25}$

Practice

Time to practice! We will be moving on to the last topic in the next article!