How can I take a partial derivative of an equation like $2u+v=5$?

Question

I know that $\frac{du}{dx} = 0$ and $\frac{du}{dy} = 0$. But I don't undestand how? how can you take a derivative of this when you don't know the value of either u or v?

Answer 1

When given an equation like this, you can apply implicit differentiation, which means that you differentiate each side of the equations as though they're actually a function that hasn't been explicitly stated. And then, since they're supposed to represent the same function, their derivatives must also be equal.

So it looks something like this:

$$\begin{eqnarray}2u + v & = & 5 \\ \frac{\partial}{\partial x} \left(2u + v \right) & = & \frac{\partial}{\partial x}(5) \\ \frac{\partial}{\partial x}(2u) + \frac{\partial}{\partial x}(v) & = & 0 \mbox{ (derivative of a constant is 0)} \\ 2\frac{\partial u}{\partial x} + \frac{\partial v}{\partial x} & = & 0 \end{eqnarray}$$

and the same thing happens for $\frac{\partial}{\partial y}$.

It's equivalent to introducing the phantom function $f(x, y) = 2u + v$, finding $\frac{\partial f}{\partial x}$, then working with the equation $f(x, y) = 5$, but since we'll be discarding $f$ anyway we just never state it explicitly.