What is differentiation and how is it done?

Differentation is a type of calculus that allows us to work out the rate of change. For example, if we have a straight line graph such as y=2x, we know that the gradient of the line is 2. If we take a curve such as y=x²+2x+9, the gradient is different at different points on the curve and so we use differentation to work this out. 

To carry out differentation you must use the following rule:

Bring the power down to the front and reduce the power by 1.

So how does this rule work in practice?

If we have an equation f(x), after we differentiate it we call it a derivative and use the notation f'(x).

So if f(x)=x^a, then f'(x)=ax^a-1. So we have brought the power a down to the front of x and then reduced a by 1. 

Now let's take a numerical example:

f(x)=x² so then f'(x)=2x

If we have an equation like f(x)=x³+2x²+9x+4 we differentiate each term separately.

The derivative of x³ is 3x²

The derivative of 2x² is 4x

The derivative of 9x is just 9 as the power reduces to 0

The derivative of 4 is 0. Any number that is not associated with a variable (such as x) will differentiate to 0. 

So f'(x)=3x²+4x+9.

If we wanted to find the gradient of f(x)=x³+2x²+9x+4 at x=2, we differentiate the equation (as done above) and then substitute the appropirate value of x. 

 We have f'(x)=3x²+4x+9, and then substitute x=2.

So the gradient of the curve at x=2 is 3(2²)+4(2)+9=29.