An Introduction To Differentiation

MathsDirect

Differentiation is the maths of changing variables. In science, what is really important, is not what value something has at the moment, but what value it will have. This is where differentiation comes in. By differentiating a variable, we find the rate at which it is changing. To begin with, we consider the gradients of curves, but soon move on, to apply differentiation to quantities changing with time.

The Gradient Of A Curve

For a straight line, we defined the gradient as

This gradient was constant

If our graph is curved however, then the gradient will constantly be changing.

The curves on the right are the same, but you can see that their gradients at the two points are very different.

We define the gradient of a curve at a point, as being the gradient of the tangent to the curve at that point.

Clearly, this will change, depending on where you are on the curve. In other words, the gradient will be a function of x.

There is a special notation for differentiation

If the curve has an equation connecting y & x
then the gradient is written

There are simple rules to work out the gradient functions of different curves. You do not need to know how these rules are arrived at. It might, however be of interest to you.

If you want to see how the general rules are derived, click on first principles.

If not, click on general formula.