The method of finding the derivative of a function is called differentiation. In this section, we’ll see how the definition of the derivative can be used to find the derivative of different functions. Later on, once you are more comfortable with the definition, you can use the defined rules to differentiate a function.
Let’s start with a simple example of a linear function m(x) = 2x+5. We can see that m(x) changes at a constant rate. We can differentiate this function as follows.
Derivative of m(x) = 2x+5
The above figure shows how the function m(x) is changing and it also shows that no matter which value of x, we choose the rate of change of m(x) always remains a 2.
Suppose we have the function g(x) given by: g(x) = x^2. The figure below shows how the derivative of g(x) w.r.t. x is calculated. There is also a plot of the function and its derivative in the figure.
Derivative of g(x) = x^2
As g’(x) = 2x, hence g’(0) = 0, g’(1) = 2, g’(2) = 4 and g’(-1) = -2, g’(-2) = -4
From the figure, we can see that the value of g(x) is very large for large negative values of x. When x < 0, increasing x decreases g(x) and hence g’(x) < 0 for x<0. The graph flattens out for x=0, where the derivative or rate of change of g(x) becomes zero. When x>0, g(x) increases quadratically with the increase in x, and hence, the derivative is also positive.
Suppose we have the function h(x) = 1/x. Shown below is the differentiation of h(x) w.r.t. x (for x ≠0) and the figure illustrating the derivative. The blue curve denotes h(x) and the red curve its corresponding derivative.