Find the derivative of \(y\) with respect to \(x\)

Solution

\[\begin{aligned}
&\mathrm{d}(x \sin (a+y)+\sin a \cos (a+y))=\mathrm{d}(0)=0 \\\\
&\mathrm{~d}(x \sin (a+y))+\sin a d(\cos (a+y))=0 \\\\
&\sin (a+y) d x+x \mathrm{~d}(\sin (a+y))+\sin a \mathrm{~d}(\cos (a+y))=0
\end{aligned}\]

\[\begin{aligned}
&\sin (a+y) d x+x(\cos (a+y)) d y-\sin a \sin (a+y) d y=0 \\\\
&\sin (a+y) d x+(x \cos (a+y)-\sin a \sin (a+y)) d y=0 \\\\
&\Rightarrow \frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{\sin (a+y)}{\sin a \sin (a+y)-x \cos (a+y)} \\\\
&=\frac{\sin (a+y)}{\sin a \sin (a+y)+\cos (a+y)\left(\frac{\sin a \cos (a+y)}{\sin (a+y)}\right)} \\\\
&=\frac{\sin (a+y)}{\sin a \sin (a+y)+\left(\frac{\sin a \cos ^{2}(a+y)}{\sin (a+y)}\right)}
\end{aligned}\]