What is the two-sided derivation of implicit function, the simplest x ^ 2 = 1-y ^ 2 The simplest x ^ 2 = 1-y ^ 2 knows how to find the derivative on the left, but the right doesn't understand. How can there be y, y '? And LN y = xlnx

What is the two-sided derivation of implicit function, the simplest x ^ 2 = 1-y ^ 2 The simplest x ^ 2 = 1-y ^ 2 knows how to find the derivative on the left, but the right doesn't understand. How can there be y, y '? And LN y = xlnx

Implicit function derivation is similar to high school compound function!
What is hidden in LNY = xlnx is y = f (x) (determined by the corresponding equation), so the left side is equivalent to (LNF (x)) "= 1 / F (x) * f (x)"
In this way, we get 1 / y * y "= LNX + 1 and Y" = (LNX + 1) / y

Implicit function derivative, both sides at the same time to x derivative what does it mean? Seek detailed explanation

By substituting the implicit function y = y (x) into the equation, an identity is obtained, so the two sides are still identical after derivation
On the left side of the equation is the function of X, so the derivative of X. The derivative of e ^ y to X is the derivative of a composite function, and Y is the intermediate variable. The remaining derivatives of XY and E are simple

How to deal with the constant in the derivation of implicit function? For example, if e ^ y + xy-e = 0, the left derivative becomes e ^ y '+ y + x y', how to get x ^ y '

For e ^ y + xy-e = 0,
E ^ y * y '(derivation rule of composite function)
Y + X * y '(derivation of multiplication of two functions: leading x to 1, multiplying with y, deriving y again, multiplying with X, adding two terms)