Question

If $3\sin \left(xy\right)+4\cos \left(xy\right)=5$, then $\frac{dy}{dx}=$

• A. $\frac{-y}{x}$
• B. $\frac{y}{x}$
• C. $\frac{x}{y}$
• D. none of these

Answer 1

The correct option is A

$\frac{-y}{x}$

Explanation for correct option:

Finding the value of $\frac{\mathbf{d}\mathbf{y}}{\mathbf{d}\mathbf{x}\mathbf{}}\mathbf{}\mathbf{}\mathbf{:}$
Given $3\sin \left(xy\right)+4\cos \left(xy\right)=5$

Differentiate with respect to $x$

$\begin{array}{rcl}3\cos xy\left(\frac{d}{dx}\right)\left(xy\right)–4\sin \left(xy\right)\left(\frac{d}{dx}\right)\left(xy\right)& =& 0\end{array}$

$\Rightarrow$ $\begin{array}{rcl}[3\cos xy–4\sin (xy)]\left(\frac{d}{dx}\right)\left(xy\right)& =& 0\end{array}$

$\Rightarrow$ $\begin{array}{rcl}[3\cos xy–4\sin (xy)](x\frac{dy}{dx}+y)& =& 0\end{array}$

$\Rightarrow$ $\begin{array}{rcl}\left(\frac{dy}{dx}\right)x[3\cos xy–4\sin (xy)]& =& -y[3\cos xy–4\sin (xy)]\end{array}$

$\Rightarrow$ $\begin{array}{rcl}\left(\frac{dy}{dx}\right)& =& \frac{-y[3\cos xy–4\sin (xy)]}{x[3\cos xy–4\sin (xy)]}\\ & =& \frac{-y}{x}\end{array}$

Hence, option (A) is the correct answer.