Question

Can someone please help me with this....differentiate,, y=(cos(x)^x

Answer 1

$y=\cos^xx=e^{\ln\cos^xx}=e^{x\ln\cos x}$

Now differentiate, applying the chain rule on the right hand side.

$\dfrac{\mathrm dy}{\mathrm dx}=e^{x\ln\cos x}\dfrac{\mathrm d}{\mathrm dx}[x\ln\cos x]$
$\dfrac{\mathrm dy}{\mathrm dx}=e^{x\ln\cos x}\left(\ln\cos x-\dfrac{x\sin x}{\cos x}\right)$
$\dfrac{\mathrm dy}{\mathrm dx}=e^{x\ln\cos x}\left(\ln\cos x-x\tan x\right)$
$\dfrac{\mathrm dy}{\mathrm dx}=\cos^xx\left(\ln\cos x-x\tan x\right)$