If ${\mathrm{x}}^{\mathrm{m}}{\mathrm{y}}^{\mathrm{n}}={\left(\mathrm{x}+\mathrm{y}\right)}^{\mathrm{m}+\mathrm{n}}$ then increasing order of the value of  $\frac{dy}{dx}$ at 

$A) \left(1,2\right)$

$B) \left(2,6\right)$

$C)\left(5,5\right)$

$D) \left(5,-15\right)$

 

The given equation is $x^m{y}^n=x^{m+n}$

apply logarithm on both sides

$m\log x+n\log y=\left(m+n\right)\log x$

differentiate both sides

$\begin{array}{rcl}\frac{m}{x}+\frac{n}{y}\frac{dy}{dx}& =& \frac{m+n}{x}\\ \frac{dy}{dx}& =& \frac{n}{x}·\frac{y}{n}\\ & =& \frac{y}{x}\end{array}$

Substitute the following points

$A) \left(1,2\right)$, the value of $\frac{dy}{dx}$ is $2$

$B) \left(2,6\right)$the value of $\frac{dy}{dx}$ is $3$

$C)\left(5,5\right)$the value of $\frac{dy}{dx}$ is $1$

$D) \left(5,-15\right)$the value of $\frac{dy}{dx}$ is $-3$

The above values in increasing orer is $D,C,A,B$