Is this a total derivative - if yes, why minus This is the beginning of the derivation of equation of motion from the Solow model (Romer, 2019):

My question is: is this the total derivative (w.r.t. ultimate source of change &quot;t&quot;)? If yes, why there is a minus sign in front of the 2nd and 3rd term?
I assume because they are in the denominator. But I was not able find any rules for this on net. Everywhere are only &quot;+&quot; symbols. Thanks.
This is how I did it. Is it wrong? I did total derivative and got the same result as in the textbook:
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)