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.

Nth Derivative - dCode

Tag(s) : Functions

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

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.

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.

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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