I would like to know how to differentiate the following function: $f(t)=e^{{-5}(t-2)}$. I think this function exists at $t=0$, but the derivative is valid only for $t>0$ or $t=0^+$.
$$f'(t)=-5e^{{-5}(t)}$$ Is this correct? How would I express this better? I'm a M.S.E.E., not an applied mathematician. This function describes capacitor current after a sudden change of voltage. Thanks. Fred
To compute the derivative of any exponential function, use the following formula:
$$\dfrac{d}{dx}\left(a^x\right)=\left(a^x\right)\left(\ln{a}\right)\left(\dfrac{d}{dx}(x)\right)$$
Hence,
$$\dfrac{d}{dt}\left(e^{-5(t-2)}\right)=\left(e^{-5(t-2)}\right)\underbrace{\left(\ln{e}\right)}_{= \ 1}\underbrace{\left(\dfrac{d}{dt}(-5(t-2))\right)}_{= - 5}$$
$$=\boxed{-5e^{-5(t-2)}}$$