Funktion $$$\sin{\left(x \right)} - \cos{\left(x \right)}$$$ derivaatta
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)$$$.
Ratkaisu
Summan/erotuksen derivaatta on derivaattojen summa/erotus:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right) - \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$Kosinin derivaatta on $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right) = - {\color{red}\left(- \sin{\left(x \right)}\right)} + \frac{d}{dx} \left(\sin{\left(x \right)}\right)$$Sinin derivaatta on $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$\sin{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} = \sin{\left(x \right)} + {\color{red}\left(\cos{\left(x \right)}\right)}$$Sievennä:
$$\sin{\left(x \right)} + \cos{\left(x \right)} = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$Näin ollen, $$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$$.
Vastaus
$$$\frac{d}{dx} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) = \sqrt{2} \sin{\left(x + \frac{\pi}{4} \right)}$$$A