Question

If $xy={\tan }^{-1}\left(xy\right)+co{t}^{-1}\left(xy\right)$, then $\frac{dy}{dx}$ is equal to

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

Answer 1

The correct option is B

$-\frac{y}{x}$

Explanation for the correct option.

We know that ${\tan }^{-1}x+co{t}^{-1}x=\frac{\mathrm{\pi }}{2}$.

So, $xy={\tan }^{-1}\left(xy\right)+co{t}^{-1}\left(xy\right)$ can be written as

$xy=\frac{\mathrm{\pi }}{2}$

By differentiating with respect to $x$, we get

$\begin{array}{rcl}x\frac{dy}{dx}+1y& =& 0\left[\mathrm{by}\mathrm{product}\mathrm{rule}\right]\\ \Rightarrow x\frac{dy}{dx}& =& -y\\ \Rightarrow \frac{dy}{dx}& =& -\frac{y}{x}\end{array}$

Hence, option B is correct.