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.

Tangent line of a function

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.

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