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I was trying to understand why the chain rule is needed to differentiate expressions implicitly.

I began by analyzing the equation used by most websites I visited:

e.g. x^{2}+y^{2}= 10

After a lot of thinking, I got to a reasoning that satisfied me... Here it goes:

From my understanding, the variable y is a function of x. This function of x is being squared. This means that we can think of f(x) as part of another function (e.g. u = g(y) = y^2). Hence, y^2 is a composite function and, thus, differentiating it would require the chain rule.

However, after coming across some different type of questions I am no longer sure my train of thought is valid. For example:

6x^2+17y = 0.

I have read that to differentiate 17 y with respect to x we also have to apply the chain rule. This does not fit with my original reasoning (since, to my eyes, y cannot be thought of as a composite function in this case)

Can anyone please help me understand why we have to use the chain rule to differentiatie implicitly?

Thank you in advance!

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# Implicit Differentiation and the Chain Rule

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