Question

How do you find the derivative of `x+xy=y^2`?

Answer 1

I assume that you want the derivative of `y` with respect to `x`. I get `dy/dx = (1+y)/(2y-x)`

Explanation:

There is no such thing as the derivative of an equation. We can find the derivative of either variable with respect to a variable. I assume that you want to find `dy/dx`.

`x+xy=y^2`

We assume that `y` is some function of `x` that we haven't found. Think of it as "some stuff in parentheses".

`underbrace(x)_"term 1"+underbrace(x("some stuff"))_"term 2" = underbrace(("some stuff")^2)_"term 3"`

We have 3 terms and we will differentiate term-by-term.

In order to differentiate `x("some stuff")`, we'll need the product rule and for `("some stuff")^2` we'll need the power rule and the chain rule.

`d/dx(x) + d/dx(xy)=d/dx(y^2)`

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

`1+y+xdy/dx = 2y dy/dx`

There are 4 terms now. Two of them include a factor of `dy/dx` and the other two do not. Solve for `dy/dx`.

`1+y = 2y dy/dx - x dy/dx`

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

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