What is this question referring to?

[Needhelp2013]

Homework Statement

What is this question en example of:
Given u=lnsin(xy) show that

Homework Equations

du=cot(xy)(ydx + xdy)

The Attempt at a Solution

I think it is total differentiation.

 

[Simon Bridge]

I'd call it implicit differentiation: ie.
if $u=xy$, then differentiating everything wrt x tells me $$\frac{du}{dx}=y+x\frac{dy}{dx} \Rightarrow du = ydx+xdy$$

 

[CompuChip]

This is a convenient shortcut, but remember that $\frac{du}{dx}$ is not really a fraction, although it looks like one.

Officially the definition is
$$du = \frac{\partial u}{\partial x} \, dx + \frac{\partial u}{\partial y} \, dy$$
for u = u(x, y).
That result du = y dx + x dy in Simon's example comes out correctly by "multiplying through by dx" is not completely co-incidental, but you should realize there's no real multiplication involved as "dx" is not really a number.