Calculus

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Can someone explain the method of implicit differentiation to me.

I know that we differentiation both sides of the equation with respect to x (I don't understand what they mean by with respect to x). And then y is a function of x...I don't understand what that means.

For example,
Find y'' if x^4+y^4 = 16

Basically we need to differentiate both sides..as far as I know..

so ..(for y')
4x^3+4y^3 = 0

BUT..
in the textbook it's

4x^3+4y^3y' = 0

Where does the y' come from?

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