Question

If $x{e}^{xy}-y-{\sin }^{2}x=0$ then $\frac{dy}{dx}$ at $x=0$ is

Answer 1

At $x=0$
$0-y-{\sin }^{2}0=0\Rightarrow y=0$
$x{e}^{xy}-y-{\sin }^{2}x=0$
Differentiating w.r.t. $x$,
$\Rightarrow e^{xy}+x{e}^{xy}(x{y}^{\prime }+y)-y^{\prime }-2\sin x\cos x=0$
Putting $x=0$
$\Rightarrow e^0+0-y^{\prime }-0=0\Rightarrow y^{\prime }=1$