Deriving Simple Tax Multiplier with Calculus (Macro Ugrad)
So here $C$ is a function with the argument $Y-T$, like $C=f(Y-T)$

So differentiate the equation: $dY=C' * (dY-dT)$

The result immediately follows like in the book. 

The reason you don't do the partial derivative is that C has only one argument - while Y and T are in there, it only takes in 1 input.