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Second Derivative

Tool for Second Derivative calculation f''. The second derivative is the application of the derivation tool to the (first) derivative of a function, a double derivation on the same variable.

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Second Derivative -

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# Second Derivative

## Second Derivative Calculator

### What is a second derivative? (Definition)

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

### How to calculate a second derivative?

Calculate the derivative of the function, then the derivative of the derivative.

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

### How to write a second derivative?

A second derivative can be written $f´´(x)$ or $f^{(2)}(x)$ or $\ddot{f}$ (double dot) or $\frac{d^2f}{dx^2}$.

On dCode use f ' ' which is the most used notation (and the fastest to write).

### How to use the second derivative for a monotony table?

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

### Which functions do not have a second order derivative?

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 differentiation/derivative domain of a function calculator.

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Second Derivative on dCode.fr [online website], retrieved on 2023-02-08, https://www.dcode.fr/second-derivative

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