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Andra derivatan av $$$\cosh{\left(x \right)}$$$
Relaterade kalkylatorer: Derivata-beräknare, Kalkylator för logaritmisk derivering
Din inmatning
Bestäm $$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right)$$$.
Lösning
Bestäm den första derivatan $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right)$$$
Derivatan av hyperbolisk cosinus är $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cosh{\left(x \right)}\right)\right)} = {\color{red}\left(\sinh{\left(x \right)}\right)}$$Alltså, $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$.
Därefter, $$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \frac{d}{dx} \left(\sinh{\left(x \right)}\right)$$$
Derivatan av hyperbolisk sinus är $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sinh{\left(x \right)}\right)\right)} = {\color{red}\left(\cosh{\left(x \right)}\right)}$$Alltså, $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$.
Således, $$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$.
Svar
$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$A