Derivatives of equations Assume that $x$ and $y$ are related by the equation $y\ln x=e^{1−x}+y^3$. Compute $dy/dx$ evaluated at $x=1$.

I do not understand how to compute the derivative of an equation. Please explain.
Here are the first few steps you need to do. As others have noted, it is called implicit differentiation. It is very similar to "normal" differentiation, but the difference is that you consider $y$ to be a function of $x$. When you want to differentiate it with respect to $x$, you get that $\frac{d}{dx}(y) = \frac{dy}{dx}$ (very simply) . Also, you need to remember to apply to $y$ all of the rules you normally apply to functions of $x$. For example, you need to apply the product rule to $\frac{d}{dx}(xy)$, among other things.

So you have that

$$y\ln x = e^{1-x} + y^3$$
thus
$$\frac{d}{dx} (y\ln x) =  \frac{d}{dx} ( e^{1-x} + y^3 )$$
so
$$\frac{dy}{dx}\ln x + \frac{y}{x} = -e^{1-x} + 3y^2 \frac{dy}{dx}$$

Now you should be able to solve for $\frac{dy}{dx}$, which will be dependent of both $x$ and $y$.
Hint: You're not computing the derivative of an equation. You're computing the derivative of $y$ with respect to $x$. Since you aren't given $y$ in terms of $x$, your answer will most likely be in terms of both $y$ and $x$, not just $x$.