Second Derivative

The second derivative is the application of the derivative on the first derivative of a function.

Example: $$ f(x) = x^2+\sin(x) \Rightarrow f'(x) = 2 x+\cos(x) \Rightarrow f''(x) = 2 - \sin(x) $$

A second derivative can be written \( f'' (x) \) or \( f^{(2)} (x) \) or \( \frac{d^2f}{dx^2} \). On dCode prefer f ' ' which is the most used notation.

The second derivative is used to know the variation of the slope of the curve representing the function. For a given interval:

- a positive second derivative means an increase of the slope (convex function)

- a negative second derivative signifies a decrease in thought (concave function)

- a zero second derivative means a straight / straight curve

For a given point:

- a second derivative canceling with a change of sign means a point of inflection, the curvature of the graphical representation changes and is reversed. It can be a maximum of the function or a minimum of the function.

Any function that is non-continuous, and / or non-differentiable in at least one point, does not have a second derivative. See the tools definition domains of a function, and the derivative domain of a function.