Understanding derivatives of exponential growth functions

Question

If I have an equation such as $$P = P_0e^{-0.7t}$$

If I need to find the derivative of P, how would I go about that? I tried to test with a math application called Maple, but I get an error that the function is recursive.

Would I start with that hint, that it is recursive and change P to $P_0e^{-0.7t}$? Making the equation look like this?

$$P_0e^{-0.7t} = P_0e^{-0.7t}$$

lol that just seems ridiculous.

Answer 1

I assume you want to look at the equation$$P=P_0 \cdot e^{-0.7t}.$$ For that use a subscript (_) between $P$ and $0$, i.e. write $P\_0$ to get $P_0$. I assume that's the problem, maple interprets your equation as $P=P\cdot 0 \cdot e^{-0.7t}$.
The derivative on the other hand is trivial if you know the derivative of the exponential $t \mapsto e^t$ and the chain rule, the we have $$P' = -0.7 \cdot P_0 e^{-0.7 t}.$$

Answer 2

I assume $P_0$ is the initial condition and for the function $P(t) = P_0 e^{-0,7t}$ it will only make sense to differentiate $P(t)$ with respect to $t$: $$ \frac{d}{dt} P(t) = -0.7P_0 e^{-0,7t}. $$