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:

chains

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:

proof-cr

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:

example-cr

…To be continued

dun dun dunnnnyuunjjnnnn

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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:

product-rule

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:

product-rule

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

picture1

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.

maximum

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.

minimum

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.

inflection

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:

funct1

Here it is shown on a graph:

sp1

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.

SP2.png

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.

funct1

funct2

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

graph_20161220_040635

Now lets compare both of the functions together:

sp4

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.

funct2

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

quad1

Therefore at the stationary points, x equals:

quad2

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

funct1

stat1

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

stat2

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:

2nddr

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

funct1

funct2

2nddrrr

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

 

drrrrrrrrrrrx

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.