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Funktion $$$\cosh{\left(x \right)}$$$ toinen derivaatta
Aiheeseen liittyvät laskurit: Derivointilaskin, Logaritmisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right)$$$.
Ratkaisu
Laske ensimmäinen derivaatta $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right)$$$
Hyperbolisen kosinin derivaatta on $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cosh{\left(x \right)}\right)\right)} = {\color{red}\left(\sinh{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(\cosh{\left(x \right)}\right) = \sinh{\left(x \right)}$$$.
Seuraavaksi $$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \frac{d}{dx} \left(\sinh{\left(x \right)}\right)$$$
Hyperbolisen sinin derivaatta on $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sinh{\left(x \right)}\right)\right)} = {\color{red}\left(\cosh{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(\sinh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$.
Siispä $$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$.
Vastaus
$$$\frac{d^{2}}{dx^{2}} \left(\cosh{\left(x \right)}\right) = \cosh{\left(x \right)}$$$A