Trouble at differentiating a consumption function
This does not seem to have anything to do with calculus.
The idea is that the income not consumed $Y_d - C$ is saved (usually denoted by $S$).
This saving is then lent out to companies (via banks) who invest it (usually denoted by $I$), and the accumulated capital is used in production. I am guessing this is denoted by $A$?
And the change of $A$ in time is the investment (assuming there is no amortization).
Hence
$$
Y_d - C = S = I = \frac{dA}{dt}
$$