Introduction to Differential Calculus

First Principles

Calculus concerns two main processes. Differentiation and Integration.

Differentiation is the examination of the gradient of a function at certain points, and as we have already seen, this must be done using the derivative of the function.
Integration is finding the area bounded by a function. (This guide only concerns differentiation)|

During the 17th Century, Sir Isaac Newton in England and Gottfried Leibniz in Germany had independently formalized the method of calculus. Their methods were based on the discovery that differentiation and integration were closely related. This became the 'Fundamental Theorem of Calculus'.

To properly introduce differential calculus and to properly prove how the derivative is found, we must understand the First Principles.

First Principles defines the derivative as a limit. Usually, we imagine that the gradient of a line can only be found if we know two points right? Well then, if the tangent only touches the function at one point, how do we know its gradient?

This is where the limiting process comes in. Two points are actually used to find the gradient of a tangent. The limiting process diminishes the distance between these two points so that it becomes one point. Using this method, the gradient of the tangent at the point can be found.

Let's take a look at the equation of the general function,  f(x). The equation y=f(x) creates the line on the graph.

Our goal is to find the gradient at the point (x, f(x)).
On the x-axis, there is a point labeled x, and another point, which has a distance of 'h' units away. Thus this other point is labeled x + h.
The vertical distance from the x-axis to the line would be the y value.

Since y = f(x), the vertical distance from x to the line is f(x). The same applies to x + h, its vertical distance to the line being f(x+h).

Now we have two points, and that means we can find the gradient of the line between the two points (in red). Recalling from basic linear equations, the gradient of a line is equal to the rise over the run. Rise being changed in the y - value, whilst run being the change in the x - value.

In this case, our rise is given by f(x+h) - f(x) and our run, is given by h.
Therefore, our gradient of the red line is given by,