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Differentiation is a tool of mathematics that is primarily used for calculating rates of change.
In Mechanics, the rate of change of displacement (with respect to time) is the velocity and the rate of change of velocity (with respect to time) is the acceleration.
When illustrating a function on a graph the rate of change is represented by the gradient.
This means that the main aim of differentiation is to find the gradient at a specific point on a graph.
Very often we know the equation for a graph (e.g. y = 3x2). The graph of this equation looks like this:
As you can see, the gradients at the points a, b and c are quite different...
At GCSE level we estimate the gradient of a curve by drawing a tangent. Differentiation allows us to find the exact gradient at any of these points.
Imagine two points on any curve where x is a function of y (written y = f(x)):
If we connect points b and a with a straight line, sometimes called a chord (as shown by the dotted line) then the gradient of this line is a very rough approximation to the gradient of the curve.
If the co-ordinates of b are (x1, y1) and the co-ordinates of a are (x2, y2) then the gradient of the line
The closer the points a and b are to each other, the more accurately we can measure the gradient of the line.
Using the graph above, but making the gap between a and b minimal, we can re-write our gradient:
You can express the position of a by saying it is the position of b plus an extra δx along the x-axis and an extra δy along the y-axis,
i.e. a = (x1 + δx , f (x1 + δx))
As a moves closer to b, δx becomes smaller and the chord ab becomes the tangent to the line at a (and therefore an exact measurement of the gradient of the line at a).
and that the correct value for the gradient occurs as δx → 0, we get the formula:
This idea is differentiation from first principles and is useful to know, though the rules that result from this are the skills that are normally tested at A level.
An example of differentiation from first principles
Find the gradient of the function y = 3x2 when x = 5.
when x = 5,
So the gradient of the function y = 3x2 is 30 at (5, 75).