Second Derivative

Question 1

How to calculate a second derivative?

Answer

The second derivative (or second order derivative) is the application of the derivative on the first derivative of a function.

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

In physics the second derivative is usually used for acceleration calculations

Question 2

How to write a second derivative?

Answer

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 (and the fastest to write).

Question 3

How to use the second derivative for a monotony table?

Answer

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 is a stationnary point that can be a maximum of the function or a minimum of the function.

Question 4

Which functions do not have a second order derivative?

Answer

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.

Second Derivative