What actully is the derivative of a function? What does it represent?
The derivative of a function is simply the gradient of that specific function at a general point, x. However, to actually understand why it is the case that 2x is the derivative of x2, and not just by using a rule, we must have a think about gradients.(Using illustrations to aid explanation)As it is known from younger years of study, the gradient of a straight line is calculated by finding how much the line rises over a period of x values, otherwise commonly known as the rise over the run. However, when calculating the gradient of a non-straight line, this methd cannot be used in order to find the gradient. Therefore, in order to find an approximate value of the gradient of the non-straight line, two points on the line can be chosen and a straight can be drawn against this, and used against as an approximate. As the distance between these points on the line decreases, the accuracy of the approximate increases.This therefore means that the gradient of any line can be found by the value of the rise of the line over a very small distance dx, as the value of dx tends to zero. This leads to the generation of th differentiation rules you will have seen before (shown in boardwork)
