Funktion $$$\ln^{2}\left(u\right)$$$ derivaatta
Aiheeseen liittyvät laskurit: Logaritmisen derivoinnin laskin, Vaiheittainen implisiittisen derivoinnin laskin
Syötteesi
Määritä $$$\frac{d}{du} \left(\ln^{2}\left(u\right)\right)$$$.
Ratkaisu
Funktio $$$\ln^{2}\left(u\right)$$$ on kahden funktion $$$f{\left(v \right)} = v^{2}$$$ ja $$$g{\left(u \right)} = \ln\left(u\right)$$$ yhdistelmä $$$f{\left(g{\left(u \right)} \right)}$$$.
Sovella ketjusääntöä $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln^{2}\left(u\right)\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(v^{2}\right) \frac{d}{du} \left(\ln\left(u\right)\right)\right)}$$Sovella potenssisääntöä $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$, kun $$$n = 2$$$:
$${\color{red}\left(\frac{d}{dv} \left(v^{2}\right)\right)} \frac{d}{du} \left(\ln\left(u\right)\right) = {\color{red}\left(2 v\right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$Palaa alkuperäiseen muuttujaan:
$$2 {\color{red}\left(v\right)} \frac{d}{du} \left(\ln\left(u\right)\right) = 2 {\color{red}\left(\ln\left(u\right)\right)} \frac{d}{du} \left(\ln\left(u\right)\right)$$Luonnollisen logaritmin derivaatta on $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$$2 \ln\left(u\right) {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} = 2 \ln\left(u\right) {\color{red}\left(\frac{1}{u}\right)}$$Näin ollen, $$$\frac{d}{du} \left(\ln^{2}\left(u\right)\right) = \frac{2 \ln\left(u\right)}{u}$$$.
Vastaus
$$$\frac{d}{du} \left(\ln^{2}\left(u\right)\right) = \frac{2 \ln\left(u\right)}{u}$$$A