I don't understand differentiation. How does it work?
The first thing to know is that it is absolutely fine to be struggling with differentiation. It is one of the harder topics of A Level maths that you won't have seen before. I know it certainly took me a while to get to grips with.
Before we look at how to differentiate, let’s briefly go over why we would want to differentiate something in the first place. Differentiation is all about rates of change – how quickly something increases or decreases. For example, if the local council can work out by how much the population of the area is increasing, they can calculate how many houses they should build over the coming years. Can you think of any other examples where finding the rate of change is useful?
In the case of functions of the form y = f(x), differentiating with respect to x means finding the rate of change of y as we increase x. Does y increase, decrease, or stay the same? And if it increases or decreases, how much does it increase or decrease? What we’re looking for is the gradient.So, what is the derivative of y = 3x – 1? This is a straight line. The rate of change of y as x increases is constant. What we’re looking for is how much y changes when x increases by 1. This is +3, and this is constant for all points along the straight line.
Now let’s look at a slightly more complicated function like y = f(x), where f(x) = x2. We can see that the rate of change of y is not the same for all values of x. For x < 0, y is decreasing, whereas for x > 0, y is increasing. So, what is the derivative of y with respect to x at x = 3? We want the gradient of the function at x = 3. This is the same as the gradient of the tangent line at x = 3. To see what this is, what we can do is look at the gradient of the line from the point on the curve f(x) where x = 3 to the point on the curve where x = 3 + d, and see what happens as d decreases to zero. This will give us 6. This same technique can be used at all values of x and if we do this, we find that the derivative of y = x2 is 2x.
