I'm working through a proof in my differential equations book, but I think I'm hung up on a basic calculus derivative.(adsbygoogle = window.adsbygoogle || []).push({});

If we have a function ##f(x,y)## and we substitute ##v=\frac{y}{x}## , rearrange to get ##y=vx##, and then take the derivative, supposedly by the product rule we get $$\frac{dy}{dx}=v+x\frac{dv}{dx}$$

I'm not quite sure how this works since ##v## is a function of both y and x and y itself is a function of x. What's going on here?

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# I Derivative When Substituting Variables

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