Question

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

• A. $-\cos x.e^{\sin x}+\frac{1}{x}$
• B. $-\sin x.e^{-\cos x}+\frac{1}{x}$
• C. $\sin x.e^{\cos x}+\frac{1}{x}$
• D. $-\sin x.e^{\cos x}+\frac{1}{x}$

Answer 1

The correct option is D $-\sin x.e^{\cos x}+\frac{1}{x}$

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