Differentiation

This is an extension of my earlier post on differentiation from first principles. I intend to go over the three main rules for differentiating functions, and important derivatives to know.
When you take a derivative of a function, you are essentially finding a formula for the gradient of the function (the gradient function). For example:

The gradient is clearly changing at different points, unlike linear graphs.

We can find the formula for the gradient by differentiating the equation for the graph:

the gradient function is m=2x

For the notation we can use either:

to represent the derivative of a function.

The formula above is the gradient equals two times x. So when x=0, the gradient =0; this makes sense if you look at the graph; when x=2, the gradient =4; but when x=-2, the gradient =-4. So now you know what differentiating gives us, we need to know how to do it. To differentiate a function you only need to focus on the x parts, anything else is a constant. The basic rule for polynomials of any degree:

For example to differentiate:

the number 5 is a constant in the equation above, so we ignore it

Important derivatives to know:

Now, to differentiate more complex stuff, we can use the product and quotient rule:

The product rule:

The quotient rule:

For example to differentiate:

remember the derivative of sine x is cosine x, and the above is from the quotient rule

And:

from the product rule

Finally the chain rule:

Now to use it in practise, and to differentiate the function below:

 remember that y is equal to the square root of u, and u is 16 minus x squared

 remember the square root of u is u to the power of 0.5

the equation I have derived here is the gradient function for a circle with a radius of 4

The three rules I have stated here allow you to differentiate most functions, to get the derivatives of the functions. Now we'll apply the rules on one big problem:

from the quotient rule

from the product rule

from the chain rule

I hope you have enjoyed reading-naBla5040