Definition of higher order derivatives.
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Derivatives of elementary functions and rules for derivatives. |
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For \(f: D\to\mathbb{R}\)
differentiable: \(f':
D\to\mathbb{R}\). Thus:
Definition 1. \(f:
D\to\mathbb{R}\) two times differentiable at
\(x_0\in D\), if \(f'\) differentiable in \(x_0\), i.e. if the limit \[f^{(2)}(x_0) := f''(x_0) :=
\left(f'\right)'(x_0)=\lim_{x\rightarrow
x_0}\frac{f'(x)-f'(x_0)}{x-x_0}\] exists. Then \(f''(x_0)\) second
derivative of \(f\) at \(x_0\).
Inductively, for \(n\in\mathbb{N}\):
Definition 2. \(f:
D\to\mathbb{R}\) \(n\)-times differentiable, if \(f\) is \((n-1)\)-times differentiable and \(f^{(n-1)}\) differentiable.
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