Määritä $$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right)$$$
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right)$$$.
Ratkaisu
Laske ensimmäinen derivaatta $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right)$$$
Sinin derivaatta on $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = {\color{red}\left(\cos{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$.
Seuraavaksi $$$\frac{d^{2}}{dx^{2}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(\cos{\left(x \right)}\right)$$$
Kosinin derivaatta on $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = {\color{red}\left(- \sin{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$.
Seuraavaksi $$$\frac{d^{3}}{dx^{3}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \sin{\left(x \right)}\right)$$$
Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = -1$$$ ja $$$f{\left(x \right)} = \sin{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \sin{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$Sinin derivaatta on $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = - {\color{red}\left(\cos{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(- \sin{\left(x \right)}\right) = - \cos{\left(x \right)}$$$.
Seuraavaksi $$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \frac{d}{dx} \left(- \cos{\left(x \right)}\right)$$$
Sovella vakion kerroinsääntöä $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ käyttäen $$$c = -1$$$ ja $$$f{\left(x \right)} = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(- \cos{\left(x \right)}\right)\right)} = {\color{red}\left(- \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Kosinin derivaatta on $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} = - {\color{red}\left(- \sin{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(- \cos{\left(x \right)}\right) = \sin{\left(x \right)}$$$.
Siispä $$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$.
Vastaus
$$$\frac{d^{4}}{dx^{4}} \left(\sin{\left(x \right)}\right) = \sin{\left(x \right)}$$$A