The chain rule

When trying to derive a composite function, we use the chain rule.

A composite function is a function that is comprised of two functions, one within the other. Here’s the generic formula:


This shows the derivative of y with respect to x broken down into the derivative of y with respect to u multiplied by the derivative u with respect to x.

This is proven to be the same here:


The value of you can be substituted with whatever value is needed to derive the function. Here’s and example of the chain rule being used:


…To be continued

dun dun dunnnnyuunjjnnnn


The product rule

When trying to derive a function that has two values multiplied together, we use the product rule.

This can be shown by the following formula:


This simply shows that to derive u × v, you simply derive them separately. Once you’ve done this, you multiply each of these values by the others original value, and then add them together.

Here’s a worked example:


Of course in this instance the product rule is not needed, and can also be solved the following way:


This does however prove that the product rule works, and In other instances, where the value cannot be simplified before deriving it, it is needed.

Stationary points

What is a stationary point?

Stationary points are where the gradient of a curve is equal to zero. They are divided up into three sub-groups: Maximums, Minimums, and inflection points.

At a maximum point, the value of Y is always greater than the values of Y immediately before and after its position on the X-axis.


At a minimum point, the value of Y is always less than the values of Y immediately before and after its position on the X-axis.


At an inflection point, the value of Y is either greater than the value of Y immediately before it and less than the value of Y immediately after it, or the value of Y is less than the value immediately before it and greater than the Y value immediately after it.


Finding stationary points

First of all, you must find out if there are any stationary points in the line. To do this you differentiate the function.

For this example, we’ll use the following function:


Here it is shown on a graph:


As you can see, there are two stationary points (a maximum and a minimum). Normally, there wouldn’t be a visual aid, but for the sake of this example, corresponding graphs have been included.


Using your eye, you can roughly tell that the stationary points are at -1, and 3/2 on the x-axis. 

Now let’s go back to the algebraic function, and differentiate it.



Here’s the new term shown also shown as a graph:


Now lets compare both of the functions together:


As you can see, the points at which the derived function have a Y value of 0, are also where the stationary points are on the original function. This is why the first step of finding stationary points, is finding the derivative of the function.

Now, back to the derived equation.


We must now set Y to equal zero, and then solve the quadratic.


Therefore at the stationary points, x equals:


Now we take these values, and substitute them back into the original function to find the Y coordinates:



Now we know that the stationary points on the curve are at the following coordinates:


Finding out what type of stationary point it is

Now that we know how to find the stationary points, we need to know what type of stationary point it is, without the aid of a graph.

To do this we first find the second derivative of the original function. This is represented by the following:


All this is, is the derivative of the original function, differentiated again. Here’s the example we used:




Now all we do is substitute the x values we found earlier into the formula.



If the resulting number is:  

> 0    it is a minimum.

< 0    it is a maximum.

= 0     it is an inflection point.


Therefore at ( -1 , 9 ), there is a maximum, and at ( 2/3 , -7/27 ), there is a minimum.