Question

The equation $y^2{e}^{xy}=9e^{-3}\cdot x^2$ defines $y$ as a differentiable function of $x.$ The value of $\frac{dy}{dx}$ for $x=-1$ and $y=3$ is

• A. $-\frac{15}{2}$
• B. $-\frac{9}{5}$
• C. $3$
• D. $15$

Answer 1

The correct option is D $15$

$y^2{e}^{xy}=9e^{-3}\cdot x^2$
Using product rule of differentiation,
$y^2\left(e^{xy}(x\frac{dy}{dx}+y)\right)+e^{xy}\cdot 2y\frac{dy}{dx}=9e^{-3}\cdot 2x$
Putting $x=-1$ and $y=3$
$9\left(e^{-3}(-\frac{dy}{dx}+3)\right)+e^{-3}\cdot 6\frac{dy}{dx}=-9e^{-3}\cdot 2\Rightarrow -9(\frac{dy}{dx}-3)+6\frac{dy}{dx}=-18\Rightarrow 3\frac{dy}{dx}=45\Rightarrow \frac{dy}{dx}=15$