What is Derivatives?

Derivatives are built on top of the concept of limits. They measure the difference between the values of a function in an interval whose width approaches the value zero. For example, let’s say a function f(x) is given and the goal is to calculate the derivative of that function at a point x = a using limits. It is denoted by [Tex]\frac{df}{dx}[/Tex], or f'(x). 

[Tex]\frac{df}{dx} = \lim_{h \to 0}\frac{f(x + h) – f(x)}{(x + h) – (x)} [/Tex]

At x = a,

[Tex]\frac{df}{dx} = \lim_{h \to 0}\frac{f(a + h) – f(a)}{h} [/Tex]

Notice in the figure, as the interval “h” approaches zero. The line approaches to being a tangent from a chord. This means, that now the derivative when h approaches zero, gives us the slope of the tangent at that particular point. 

Derivatives of some Basic Functions

The table below shows the derivatives of some standard basic functions. 

Derivatives are an integral part of calculus. They measure the rate of change in any quantity. Suppose there is a water tank from which water is leaking. A local engineer is asked to measure the time in which the water tank will become empty. In such a scenario, the engineer needs to know two things — the size of the water tank and the rate at which water is flowing out of it. The size of the tank can be found out easily but to measure the rate at which water is leaking he will have to use derivatives.

In this way, derivatives are intertwined in our lives. It is easy to calculate the derivatives for simple functions, but when functions become complex the correct way to approach this problem is to break the problem into subproblems that are easier to solve. Let’s see some rules and approaches to do that in the case of derivatives.