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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

## 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. It therefore measures the variation of the first derivative of the function.

### How to calculate a second derivative?

Calculate the derivative of the function(also called the first derivative), then the derivative of the derivative (called second 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, in economics it allows us to analyze phenomena linked to growth rates.

### What is the list of common second derivatives?

The second derivatives to know are:

NameFonctionSecond Derivative
constant/number$$k \in \mathbb{R}$$$$0$$
variable (only)$$x$$$$0$$
power n (exponent)$$x^n$$$$n(n-1) x^{n-2}$$
inverse$$\frac{1}{x}$$$$\frac{2}{x^3}$$
square root$$\sqrt{x}$$$$\frac{1}{4x^{3/2}}$$
natural logarithm$$\ln |x|$$$$-\frac{1}{x^2}$$
exponential$$e^x$$$$e^x$$
exponent x$$a^x$$$$a^x (\ln(a))^2$$
sine$$\sin(x)$$$$-\sin(x)$$
cosine$$\cos(x)$$$$-\cos(x)$$
tangent$$\tan(x)$$$$\frac{2\tan(x)}{\cos^2(x)}$$

### 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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