Question

If $\sin (x+y)=\log (x+y)$, then $\frac{dy}{dx}$ is equal to

• A. $2$
• B. $-2$
• C. $1$
• D. $-1$

Answer 1

The correct option is D

$-1$

Explanation for the correct option:

Differentiate both the sides of the given function with respect to x, using Chain rule that is the derivative of $f\left(g\right(x\left)\right)$ is $f\text{'}\left(g\right(x\left)\right)\cdot g\text{'}\left(x\right)$.

Given: $\sin (x+y)=\log (x+y)$,

Differentiating both the sides with respect to $x$, then

$\begin{array}{c}\cos (x+y)\left(1+\frac{dy}{dx}\right)=\frac{1}{(x+y)}\left(1+\frac{dy}{dx}\right)\\ \cos (x+y)\left(1+\frac{dy}{dx}\right)-\frac{1}{(x+y)}\left(1+\frac{dy}{dx}\right)=0\\ \left(1+\frac{dy}{dx}\right)\left[\cos (x+y)-\frac{1}{(x+y)}\right]=0\end{array}$

Which implies

$\begin{array}{c}\left(1+\frac{dy}{dx}\right)=0\\ \Rightarrow \frac{dy}{dx}=-1\end{array}$

Hence, the correct option is (D).