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