Funktion $$$1 - \tan{\left(x \right)}$$$ derivaatta
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d}{dx} \left(1 - \tan{\left(x \right)}\right)$$$.
Ratkaisu
Summan/erotuksen derivaatta on derivaattojen summa/erotus:
$${\color{red}\left(\frac{d}{dx} \left(1 - \tan{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(1\right) - \frac{d}{dx} \left(\tan{\left(x \right)}\right)\right)}$$Vakion derivaatta on $$$0$$$:
$${\color{red}\left(\frac{d}{dx} \left(1\right)\right)} - \frac{d}{dx} \left(\tan{\left(x \right)}\right) = {\color{red}\left(0\right)} - \frac{d}{dx} \left(\tan{\left(x \right)}\right)$$Tangenttifunktion derivaatta on $$$\frac{d}{dx} \left(\tan{\left(x \right)}\right) = \sec^{2}{\left(x \right)}$$$:
$$- {\color{red}\left(\frac{d}{dx} \left(\tan{\left(x \right)}\right)\right)} = - {\color{red}\left(\sec^{2}{\left(x \right)}\right)}$$Näin ollen, $$$\frac{d}{dx} \left(1 - \tan{\left(x \right)}\right) = - \sec^{2}{\left(x \right)}$$$.
Vastaus
$$$\frac{d}{dx} \left(1 - \tan{\left(x \right)}\right) = - \sec^{2}{\left(x \right)}$$$A