Derivatan av $$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$
Relaterade kalkylatorer: Kalkylator för logaritmisk derivering, Räknare för implicit derivering med steg
Din inmatning
Bestäm $$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)$$$.
Lösning
Funktionen $$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$ är sammansättningen $$$f{\left(g{\left(x \right)} \right)}$$$ av två funktioner $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ och $$$g{\left(x \right)} = \frac{2 \ln\left(x\right)}{3}$$$.
Tillämpa kedjeregeln $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$:
$${\color{red}\left(\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)}$$Derivatan av cosinus är $$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$Återgå till den ursprungliga variabeln:
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = - \sin{\left({\color{red}\left(\frac{2 \ln\left(x\right)}{3}\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$Tillämpa konstantfaktorregeln $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ med $$$c = \frac{2}{3}$$$ och $$$f{\left(x \right)} = \ln\left(x\right)$$$:
$$- \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)} = - \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{2 \frac{d}{dx} \left(\ln\left(x\right)\right)}{3}\right)}$$Derivatan av den naturliga logaritmen är $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$- \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)}}{3} = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{1}{x}\right)}}{3}$$Alltså, $$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right) = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}}{3 x}$$$.
Svar
$$$\frac{d}{dx} \left(\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right) = - \frac{2 \sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}}{3 x}$$$A