Question

If $\sin (x+y)+\cos (x+y)=\log (x+y)$ then $\frac{d^{2}y}{d{x}^2}$ is equal to

• A. $\frac{-y}{x}$
• B. $0$
• C. $-1$
• D. $1$

Answer 1

The correct option is B

$0$

Explanation for the correct answer

Step 1: Differentiate with respect to $x$ of above equation

Given the information for the solution

$\therefore \sin (x+y)+\cos (x+y)=\log (x+y)$

$\Rightarrow \cos \left(x+y\right)\left(1+\frac{dy}{dx}\right)-\sin \left(x+y\right)\left(1+\frac{dy}{dx}\right)=\left[\frac{1}{\left(x+y\right)\left(1+{\displaystyle \frac{dy}{dx}}\right)}\right]$

$\Rightarrow \left(1+\frac{dy}{dx}\right)\left[\cos \left(x+y\right)-\sin \left(x+y\right)-\left(\frac{1}{\left(x+y\right)}\right)\right]=0$

$\Rightarrow \left(1+\frac{dy}{dx}\right)=0$

$\Rightarrow \frac{dy}{dx}=-1$

Step 2: Again differentiate with respect to $x$

$\Rightarrow \frac{d^{2}y}{d{x}^2}=0$

Therefore the value of $\frac{d^{2}y}{d{x}^2}$ is equal to $0$

Hence option (B) is the correct answer.