How do you differentiate `xy = cot(xy)`?

Start off by differentiating both sides.

`d/dx (xy) = d/dx (cot(xy))`

First, let's discuss how to differentiate the left side.
We can apply the product rule.

`x(d/dx y) + y(d/dxx) = d/dx (cot(xy))`

`d/dx y = dy/dx` due to implicit differentiation and the chain rule.

So:

`x(d/dx y) + y(d/dxx) = d/dx (cot(xy))`

`xdy/dx + y = d/dx (cot(xy))`

Next, we can differentiate the right side using the chain rule.
In short terms, the chain rule is:
1) The derivative of the outer function, with the inner function plugged in...
2) ...multiplied by the derivative of the inner function.

The outer function is `cot(x)`.
The inner function is `xy`.

The derivative of the outer function is `csc^2(x)`.
Plug in the inner function and we get `csc^2(xy)`.
Then we multiply this by the derivative of the inner function.

So `xdy/dx + y = d/dx (cot(xy))` becomes:

`xdy/dx + y = csc^2(xy) * d/dx (xy)`

We've already solved `d/dx (xy)` on the left side, so we can just copy it:

`xdy/dx + y = csc^2(xy) * [xdy/dx + y]`

`xdy/dx + y =xcsc^2(xy) dy/dx + ycsc^2(xy)`

Then we need to isolate `dy/dx` so we bring all `dy/dx` terms to one side:

`xdy/dx - xcsc^2(xy) dy/dx = ycsc^2(xy) - y`

Then we factor out `dy/dx` to isolate it:

`dy/dx(x - xcsc^2(xy))= ycsc^2(xy) - y`

`dy/dx= ( ycsc^2(xy) - y)/(x - xcsc^2(xy))`