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

Answer: (D)

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