What is a derivative and how do we calculate it from first principles?

A derivative, in essence, is the gradient of a function at a point, usually denoted by dy/dx, or f'(x). It essentially means the rate of change of y with respect to x. As many functions are complex, we can no longer just draw triangles to approximately find the gradient, we must use mathematical methods to do this. As given in the A level formula book, f'(x) = lim h->0 ((f(x+h)-f(x))/h), which looks scary, but we can easily decompose what this means. If we take a function f, and two points on the x axis, say x and x+h, and corresponding points on the y axis, f(x) and f(x+h). Then finding the gradient between these points is the same method as we always have done, throughout school, change in y / change in x, giving f'(x)=(f(x+h)-f(x))/((x+h)-x), giving f'(x)=(f(x+h)-f(x))/h, however as we want the gradient at x, we must let h go towards zero, so we can determine the gradient. Thus, we write f'(x) = lim h->0 ((f(x+h)-f(x))/h), where lim h->0 means the limit as h tends to 0.
An example: Find the derivative of x².
Let f(x)=x², then f'(x)= lim h->0 ((x+h)²-x²)/h = lim h->0 ((x²+2hx+h²-x²)/h = lim h->0 (2hx+h²)/h = lim h->0 2x+h =2Which is in fact the derivate of x²