# Differentiation

Differentiation falls under the branch of mathematics called calculus. It is a measure of the rate of change of function as its input variables change. The most common example of this is your speed when travelling in a car, your speed is the rate of change of distance with respect to time or $speed=f'(x)=\frac{dx}{dt}$, where $x$ is the distance travelled, and $t$ is time. If you differentiate this function again you get the rate of change of speed with respect to time, or in other the accelleration, or even another way $accelleration = f''(x) = \frac{d^2x}{dt^2}$.

The derivative of a two dimensional function is the slope of the tangent line of the function at a specific input value. The derivative can be calculated a number of ways, including approximations using numerical methods which essentially picks two input values for the function and draws a straight line between these two points, and the slope of this line is the rate of change. By differentiating the fuction we can calculate exact rates of change, which is why a derivative is also known as the instantaneous rate of change. This value is calculated by using a mathematical concept known as limits to slope of the function at any point of the function.