Question

If $e^{xy}=y+{\sin }^{2}x$, then at $x=0$, $dy/dx$ is equal to

Answer 1

Putting $x=0$ in the given equation, we get $y=0$. Differentiating both the sides, We have
$e^{xy}+x{e}^{xy}[x\frac{dy}{dx}+y]=\frac{dy}{dx}+2\sin x\cos x$.
Putting $x=0,y=0,$ we have
$e^0+0.e^0[0\frac{dy}{dx}+0]={\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=0}+0$
$\Rightarrow$ ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=0}=1$.