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

Solution

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

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

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