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Nth Derivative

Tool for Nth Derivative calculation f^(n), so 1,2,3 or n times the application of the derivation to a function, a n-tuple iterated/successive derivation on the same variable.

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Nth Derivative -

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# Nth Derivative

## Answers to Questions (FAQ)

### How to calculate a nth derivative?

The nth derivative (or derivative of order $n$) of a function $f$ consists of the application of the derivative iteratively $n$ times on the function $f$.

Example: $$f(x) = x^4+\cos(x) \\ \Rightarrow f´(x) = 4 x^3-\sin(x) \\ \Rightarrow f´´(x) = 12x^2-\cos(x) \\ \Rightarrow f´´´(x) = 24x+\sin(x) \\ \Rightarrow f´´´´(x) = 24+\cos(x)$$

### What can be a nth derivative for?

In physics, derivatives are useful for describing systems, the first derivative of a trajectory with respect to time represents speed, the second derivative represents acceleration and the third derivative characterizes jerk.

### How to write a nth derivative?

An nth derivative can be written either $f^{(n)}(x)$ or $\frac{d^n f}{dx^n}$.

When $n$ is small (and is 1, 2 or 3), it is common to write a prime (an apostrophe) f' for the derivative, f' ' for the second derivative, f ' ' ' for the third derivative, etc.

### Which functions have remarkable successive derivatives?

The trigonometric functions $\sin$ and $\cos$ have successive periodic derivatives.

$$f^{(4n)}(x) = \cos(x) \\ f^{(4n + 1)} (x) = -\sin (x) \\ f^{(4n + 2)} (x) = -\cos (x) \\ f^{(4n + 3)} (x) = \sin (x)$$

$$f^{(4n)}(x) = \sin(x) \\ f^{(4n + 1)} (x) = \cos (x) \\ f^{(4n + 2)} (x) = -\sin (x) \\ f^{(4n + 3)} (x) = -\cos (x)$$

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