Partial derivative: confused about what equation to use 
I did get a formula for $x$ and formula for $y$ in terms of $u$ and $v$.
When finding those derivatives, which equations do I use for each? e.g. When finding $\frac{\partial u}{\partial x}$, do I use equation 1 or the equation 2? I found ($x$ in terms of $u$ and $v$).
Thanks in advance.
 A: For $du/dx, dv/dx, du/dy, dv/dy$ use the given formulae.
For the others, use the formulae you found in the first step.
To find $\dfrac{du}{dx}$, we're interested in finding how $u$ varies with respect to $x$. Since we already have an explicit relationship for $u$ in terms of $x$ and $y$, we can just differentiate that expression with respect to $x$. That is,
$$
\frac{d}{dx}u = \frac{d}{dx}(x+1)^2 + \frac{d}{dx}y^2 \\
\frac{du}{dx} = 2(x+1)\frac{dx}{dx} + 0 \\
\frac{du}{dx} = 2(x+1).
$$
If we now want to find $\dfrac{dx}{du}$, we could try and use implicit differentiation
$$
\frac{d}{du}u = \frac{d}{du}(x+1)^2 + \frac{d}{du}y^2 \\
1 = 2(x+1)\frac{dx}{du} + 2y\frac{dy}{du} \\
$$
but now the other derivative $\dfrac{dy}{du}$ is kind of tangled up in this expression, and it will be hard to separate. 
Thus, our work is made easier if we first find an explicit formula for $x$ in terms of $u$ and $v$. I think the answer you should get from the earlier part is
$$
x = \frac{1}{4}(u - v)
$$
so
$$
\frac{d}{du}x = \frac{1}{4}\left(\frac{d}{du}u - \frac{d}{du}v\right) \\
\frac{dx}{du} = \frac{1}{4}
$$
