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.


`(\mathrm dy)/(\mathrm dx)=e^(x\ln\cos x)(\mathrm d)/(\mathrm dx)[x\ln\cos x]`

`(\mathrm dy)/(\mathrm dx)=e^(x\ln\cos x)\left(\ln\cos x-(x\sin x)/(\cos x)\right)`

`(\mathrm dy)/(\mathrm dx)=e^(x\ln\cos x)\left(\ln\cos x-x\tan x\right)`

`(\mathrm dy)/(\mathrm dx)=\cos^xx\left(\ln\cos x-x\tan x\right)`