How can I take the derivative of $Q=C \cdot \Delta T$

Question

How can I take the derivative of $$Q=C \cdot \Delta T$$ with respect to time $t$?

note: $C$ is heat capacity and $T$ is temperature

Answer 1

The notations in physics books are not clear. Assume that $C$ is a constant.

If you define $Q(t)$ to be the total heat added to the object, then in fact it should be $\Delta Q=C\Delta T$. Taking derivatives gives $\frac{dQ}{dt}=C\frac{dT}{dt}$.

However, in physics books, usually $Q$ is used to mean change in internal energy of an object by heating. So better define internal energy $U$. Then $Q = \Delta U$ when $W=0$. Then it's better to write $\frac{dU}{dt}_{W=0}=C\frac{dT}{dt}$.