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Can I differentiate both sides of an equation like the one below and solve for $y$? Considering $y=f(x)$, or do I need to use implicit differentiation for something like that?

$x\cos(2x+3y)=y\sin(x)$

EDIT: answering some questions, the question is to find the derivative of f(x)

Levy Barbosa
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    What you describe is legal and IS implicit differentiation, where $y$ is 'implicitly' a function of $x$. Actually differentiating here will make the situation complicated as you can see $y$ inside $cos$ on the left and next to the $sin(x)$ on the right. An equation that looks like $y = \ldots$ will be too much to ask for. – Square Oct 11 '20 at 23:55
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    If you are trying to get an explicit formula for y, differentiating is going to increase the complexity. – The Chaz 2.0 Oct 12 '20 at 00:02

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Deriving both sides of your particular equation requires implicit differentiation, period.

Is the question find the derivative? Or is it just to solve for $y$ in terms of $x$? As others have stated, this will complicate the matter if you are trying to solve for $y$ in terms of $x$. Here's the derivative of right hand side to get you started:

$$ \frac{\partial y}{\partial x} \sin(x) + f(x) \cos(x) $$

which you can see now contains both $f(x)$ and $f'(x)$.

Charalambos Kioulos
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