I have this implicit function $$y=f(x) \iff \sin(x+y)=k \sin(x), \quad$$ where $k>1$ is a constant.
I would like to know how a small variation in $x$ propagates on $y$.
I think I need to do an implicit differentiation but then it is not so clear to me how to solve the problem.
So the derivative of the LHS is
$$\frac{\mathrm{d}}{\mathrm{d}x}(\sin(x+y)) = \cos(x+y)(1+\frac{\mathrm{d}y}{\mathrm{d}x})$$
and the derivative of the RHS is
$$\frac{\mathrm{d}}{\mathrm{d}x}(k\sin(x)) = k\cos(x)$$
And solving for $\displaystyle \frac{\mathrm{d}y}{\mathrm{d}x}$ gives
$$\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{k\cos(x)-\cos(x+y)}{\cos(x+y)}$$
and now how can I continue?
Thank you.