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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Nth Derivative
Nth Derivative Calculator
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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