1. **State the problem:** Differentiate the function $v(t) = V(1 - e^{-t/T})$ with respect to $t$. 2. **Recall the differentiation rules:** - The derivative of a constant times a function is the constant times the derivative of the function. - The derivative of $e^{u(t)}$ with respect to $t$ is $e^{u(t)} \cdot u'(t)$. 3. **Apply the derivative:** $$\frac{dv}{dt} = V \cdot \frac{d}{dt}(1 - e^{-t/T}) = V \cdot \left(0 - \frac{d}{dt} e^{-t/T}\right)$$ 4. **Differentiate the exponential:** Let $u = -\frac{t}{T}$, then $u' = -\frac{1}{T}$. $$\frac{d}{dt} e^{u} = e^{u} \cdot u' = e^{-t/T} \cdot \left(-\frac{1}{T}\right) = -\frac{1}{T} e^{-t/T}$$ 5. **Substitute back:** $$\frac{dv}{dt} = V \cdot \left(0 - \left(-\frac{1}{T} e^{-t/T}\right)\right) = V \cdot \frac{1}{T} e^{-t/T}$$ 6. **Final answer:** $$\boxed{\frac{dv}{dt} = \frac{V}{T} e^{-t/T}}$$