If$\mathit{y}{\mathit{e}}^{\mathbf{x}\mathbf{y}}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathit{x}$ find$\mathit{d}\mathit{y}\mathbf{/}\mathit{d}\mathit{x}$ and${\mathit{d}}^{\mathbf{2}}\mathit{y}\mathbf{/}\mathit{d}{\mathit{x}}^{\mathbf{2}}$ atrole="math" localid="1658830042567" $(0,0)$ .

The provided equation is $y{e}^{xy}=\sin x$.

In implicit differentiation, differentiate each side of an equation with two variables (usuallyxand y) by treating one of the variables as a function of the other.

Consider the equation below:

$y{e}^{xy}=\sin x$

Differentiate the provided equation,

$$\begin{gathered} {\mathrm{ye}}^{\mathrm{xy}}\left[\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right]+{\mathrm{e}}^{\mathrm{xy}}.\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{cosx} \\ {\mathrm{xye}}^{\mathrm{xy}}\frac{\mathrm{dy}}{\mathrm{dx}}+{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}.\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{cosx} \\ {\mathrm{xye}}^{\mathrm{xy}}\frac{\mathrm{dy}}{\mathrm{dx}}+{\mathrm{e}}^{\mathrm{xy}}.\frac{\mathrm{dy}}{\mathrm{dx}}+{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}=\mathrm{cosx} \end{gathered}$$

Since,

$$\begin{gathered} \frac{\mathrm{dy}}{\mathrm{dx}}\left[{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}\right]+{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}=\mathrm{cosx} \\ \frac{\mathrm{dy}}{\mathrm{dx}}\left[{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}\right]=\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}} \\ {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\left(0,0\right)}=\frac{\cos 0-{\left(0\right)}^2{\mathrm{e}}^0}{\left(0\right)\left(0\right){\mathrm{e}}^0+{\mathrm{e}}^0} \end{gathered}$$

Hence,

role="math" localid="1658832538492" ${\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{\left(0,0\right)}=1$

Again, differentiate the provided equation,

role="math" localid="1658831607846" $$\begin{gathered} \frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}=\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}}{{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}}\right) \\ =\left\{\frac{\left({\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}\right)\left[-\sin \mathrm{x}-\left({\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}\left(\mathrm{x}{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}+\mathrm{y}\right)+2{\mathrm{ye}}^{\mathrm{xy}}{\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}}\right)\right]-(\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}}{{\left({\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}\right)}^2}\right\} \end{gathered}$$$$\begin{gathered} \left[\left(\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right){\mathrm{e}}^{\mathrm{xy}}+{\mathrm{xye}}^{\mathrm{xy}}\left(\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right)+{\mathrm{e}}^{\mathrm{xy}}\left(\mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+\mathrm{y}\right)\right] \\ =\left\{\frac{\left({\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}\right)\left[-\mathrm{sinx}-\left({\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}\left(\mathrm{x}\left(\frac{\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}}{{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}}\right)\right)+2\mathrm{y}\right)\right]-\left(\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}\right)}{{\left({\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}\right)}^2}\right\} \end{gathered}$$

$\left[\left(\mathrm{x}\left(\frac{\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}}{{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}}\right)+\mathrm{y}\right){\mathrm{e}}^{\mathrm{xy}}+{\mathrm{xye}}^{\mathrm{xy}}\left(\mathrm{x}\left(\frac{\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}}{{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}}\right)+\mathrm{y}\right)+{\mathrm{e}}^{\mathrm{xy}}\left(\mathrm{x}\left(\frac{\mathrm{cosx}-{\mathrm{y}}^2{\mathrm{e}}^{\mathrm{xy}}}{{\mathrm{xye}}^{\mathrm{xy}}+{\mathrm{e}}^{\mathrm{xy}}}\right)+\mathrm{y}\right)\right]$

Hence,

${\left(\frac{{\mathrm{d}}^2\mathrm{y}}{{\mathrm{dx}}^2}\right)}_{\left(0,0\right)}=0$