Is this a total derivative - if yes, why minus

This is just standard derivative. The minus sign there is because you are taking derivative of a quotient and the quotient rule for derivatives is:

$$ frac{d}{dx} left(frac{f(x)}{g(x)}right) = frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}$$

In this case:

$$dot{k} = frac{d}{dt} left(frac{K(t)}{A(t)L(t)} right) = frac{dot{K}(t)A(t)L(t)-K(t)(dot{A}L(t)+A(t)dot{L}(t))}{(A(t)L(t))^2} = frac{dot{K}(t)A(t)L(t)}{(A(t)L(t))^2} - frac{K(t)(dot{A}(t)L(t)+A(t)dot{L}(t))}{(A(t)L(t))^2}= frac{dot{K}(t)}{(A(t)L(t))} - frac{K(t)(dot{A}(t)L(t)+A(t)dot{L}(t))}{(A(t)L(t))^2}$$

So you just need to remember that $A$ and $L$ are both functions of $t$ so you need to take derivative of whole qutient and quotient rule has minus there (if you want explanation for why quotient rule includes minus the right place is mathematics.se)