Second Derivative

What is the second derivative?

The second derivative is simply the derivative of the first derivative of a function. It is denoted by f'' or f⁽²⁾:

$$ f''(x) = D[ f'(x) ] $$

In other words, it is obtained by differentiating the original function f(x) twice.

A Practical Example

Let’s consider the following function:

$$ f(x) = x^2 $$

The first derivative of f(x) is:

$$ f'(x) = D[x^2] = 2x $$

The second derivative is simply the derivative of the first derivative:

$$ f''(x) = D[f'(x)] = D[2x] = 2 $$

And so on.

Note. By the same logic, the third derivative is the derivative of the second derivative - that is, the result of differentiating f(x) three times.