Last time I presented the multiplication rule of differentiation to be used when given a function that is the multiplication of two or more other functions. As you would guess, there is also a rule that handles the division of two functions.

Let’s say you have the function

\[ f( x) =\frac{x^{2}}{\text{sin}( x)}\]

This one can be solved with the multiplication rule if you remember that 1/sin(*x*) = csc(*x*). But as I haven’t told you what the derivative of csc(*x*) is, we are stuck using the following division rule. But this highlights the point that as we get deeper into maths, there are often several ways to solve a problem. The maths “arteest” is one that solves a problem elegantly.

So the following rule is the division rule. Again, I will use *u*(*x*) and *v*(*x*) to split the function up into its parts. If you have a function of the form

\[f( x) =\frac{u( x)}{v( x)}\]

then the derivative of *f*(*x*) is

\[f'( x) =\frac{u( x) v'( x) -u'( x) v( x)}{[ v( x)]^{2}}\]

As you can see, this rule is a bit more complex which is why you would use a simpler rule if possible. But it is still relatively easy to use if you keep track of which part is *u* and which part is *v*.

Using the example function above,

\[\begin{array}{{>{\displaystyle}l}} u( x) =x^{2} ,\ \ \ \ \ v( x) =\text{sin}( x)\\ u'( x) =2x,\ \ \ \ v'( x) =\text{cos}( x) \end{array}\]

So according to the division rule,

\[f'( x) =\frac{x^{2}\text{cos}( x) -2x\text{sin}( x)}{\text{sin}^{2}( x)}\]

Now you can use many rules in a single differentiation problem consider

\[f( x) =\frac{x^{2} e^{x}}{\text{sin}( x)}\]

Here, the numerator is a multiplication of two functions. So when using the division rule, you need to apply the multiplication rule for the *u*‘ part:

\[ \begin{array}{{>{\displaystyle}l}}

u( x) \ =\ x^{2} e^{x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ v( x) =\text{sin}( x)\\

u'( x) =x^{2} e^{x} +2xe^{x} \ \ \ \ \ v'( x) =\text{cos}( x)

\end{array}\]

I’ll leave it as an exercise for you to see if I correctly found *u*‘(*x*) using the multiplication rule. I used the fact that (as seen from the table I provided a couple of posts before) *eˣ* is its own derivative. Anyway, using the division rule,

\[f'( x) =\frac{x^{2} e^{x}\text{cos}( x) -\left( x^{2} e^{x} +2xe^{x}\right)\text{sin}( x)}{\text{sin}^{2}( x)}\]

So you might be thinking that you can differentiate any function as long as you know the derivatives of the individual parts. So how would you differentiate

\[ f( x) =\text{sin}\left( x^{2}\right) ?\]

This is not a multiplication of functions, but rather a function of a function. I will introduce the very powerful *chain rule* as it applies to differentiation in my next post.

*Related*