Question

If $y=(\cos x{)}^{\cos x}$, find $\frac{dy}{dx}$.

Answer 1

$y=(\cos x{)}^{\cos x}$

Taking lo logarithmic Function of base $e$ on both side

We get,

$\log y=\log \left[\right(\cos x){]}^{\cos x}$

$\log y=\cos \left[\log \right(\cos x\left)\right]$

taking differentiation on both side, we get,

$\frac{1}{y}\frac{dy}{dx}=\cos x\frac{d}{dx}\left[\log \right(\cos x\left)\right]+\frac{d}{dx}\left(\cos x\right)\left[\log \right(\cos x\left)\right]$

$\frac{1}{y}\frac{dy}{dx}=\cos -\frac{1}{\cos x}+(-\sin x)+(-\sin x)\log \left(\cos x\right)$

$\frac{dy}{dx}=-\sin xy.[1+\log (\cos x\left)\right]$

$\therefore \frac{dy}{dx}=(\cos x{)}^{\cos x}.(-\sin x).[1+\log \left(\cos x\right)]$