Question

Find derivative of $f\left(x\right)=e^{\ln x}+\sin x+8$

• A. $e^{\ln x}.\frac{1}{x^2}+\cos x$
• B. $e^{\ln x}.\frac{1}{x}-\sin x$
• C. $e^{\ln x}.\frac{1}{x}+\cos x$
• D. $e^{\ln x}.\frac{1}{x}-\cos x$

Answer 1

Step 1: Given

$y=e^{\ln x}+\sin x+8$

Step 2: Formula used

$\frac{d\left(lnx\right)}{d\left(x\right)}=\frac{1}{x}$ , $\frac{d\left(c\right)}{dx}=0$ and $\frac{d\left(e^x\right)}{dx}=e^x$ (where c is constant)

Step 2: Finding the derivative

$y^{{}^{\prime }}=\frac{d{e}^{\ln x}}{dx}+\frac{d\sin x}{dx}+\frac{d8}{dx}$
$\Rightarrow \frac{d{e}^{\ln x}}{d\ln x}\times \frac{d\ln x}{dx}+\cos x+0$
$\Rightarrow y^{{}^{\prime }}=e^{\ln x}.\frac{1}{x}+\cos x$

Hence, the correct option is D.