Quotient Rule

The quotient rule is nothing to fear.

It may sound like a difficult thing to do, but with a little bit of practice it is actually quite simple.
If you already know the product rule then you should not have a hard time learning this.
The quotient rule follows the same principles.

At this point we should know that you cannot simply take the derivative of both parts of the quotient and say that it is equal to the derivative of the entire quotient. Derivatives are not multiplicative.

So we must use the quotient rule to deal with functions that use division.
The rule is slightly more complicated than the product rule and looks like this: If we have a function f(x) = g(x)/h(x) Then f'(x) = [g'(x)h(x) - h'(x)g(x)]/[h(x)]^2 This looks fairly complicated but is actually quite easy to put into practice.

Let's take a look at an example.

Say where asked to find the derivative of f(x) = e^x/(x+3).
First we need to find the derivative of the top. From previous sections we should know that the derivative of an exponential function is simply the exponential function itself.
That is to say the derivative of e^x is e^x.

The derivative of the bottom in this case is very straightforward.
It is a simple linear function with a slope of one.
Therefore we know that the derivative of this function is simply one.

Now if we apply the quotient rule; f'(x) = [e^x * (x+3) - 1 * e^x]/(x+3)^2 f'(x) = [e^x * (x+2)]/(x+3)^2 We can further simplify this answer but it is not really necessary. We see here that we have a systematic approach to solving for this derivative. If you follow this procedure you will see that it is not that difficult to use the quotient rule. It is just another rule to remember, and as always, practice makes perfect.