Differentiate the x?! | Differentiation help needed dx/dx | Beginner question

Solution 1:

$\frac{dx}{dx} = 1$, $\frac{dy}{dy} = 1$ for any equation.

Your equation is $x = f(y) = y^2 + \frac 1y$

Note that $x$ is written in terms of $y$

You can differentiate $x$ in terms of $y$, i.e., $\frac{dx}{dy}$

$\frac{dx}{dy} = 2y -\frac{1}{y^2}$

Your claim that $\frac{dx}{dx} = 2y -\frac{1}{y^2}$ is wrong.

If you want to differentiate $y$ in terms of $x$, i.e., $\frac{dy}{dx}$

$\frac{dy}{dx} = \frac{1}{(\frac{dx}{dy})}$ , in some cases.

Else, you can write the equation as y in terms of x, and then differentiate it.