Question

How do you find `dy/dx` by implicit differentiation given `e^y=x^2+y`?

Answer 1

Note that since `y = y(x)`, taking the derivative of the entire function with respect to `y` requires you to account for that fact using the chain rule.

`(df)/(dx) = (df)/(dy)(dy)/(dx)`

where `(df)/(dy)` is the derivative of, say, `e^y` or `y` in your given function.

Thus:

`e^y (dy)/(dx) = 2x + 1(dy)/(dx)`

`=> (e^y - 1)(dy)/(dx) = 2x`

`=> color(blue)((dy)/(dx) = (2x)/(e^y - 1))`

If you know what `y` is in terms of `x`, you can plug it back in to give you a function containing just `x`.