Find derivative of $f\left(x\right)=e^{\cos x}+\ln x$

The correct option is D $-\sin x.e^{\cos x}+\frac{1}{x}$
Given: $f\left(x\right)=e^{\cos x}+\ln x$
$f^{{}^{\prime }}\left(x\right)=\frac{d{e}^{\cos x}}{dx}+\frac{d\ln x}{dx}$
$\Rightarrow \frac{d{e}^{\cos x}}{d\cos x}\times \frac{d\cos x}{dx}+\frac{1}{x}$
$\Rightarrow f^{{}^{\prime }}\left(x\right)=-\sin x.e^{\cos x}+\frac{1}{x}$