1. **State the problem:** Differentiate the function $y(t) = y_0 e^{kt}$ with respect to $t$. 2. **Recall the formula:** The derivative of an exponential function $e^{u(t)}$ with respect to $t$ is given by $$\frac{d}{dt} e^{u(t)} = e^{u(t)} \cdot \frac{du}{dt}.$$ Also, constants multiply through the derivative. 3. **Apply the formula:** Here, $u(t) = kt$, so $$\frac{du}{dt} = k.$$ 4. **Differentiate:** $$\frac{dy}{dt} = y_0 \cdot \frac{d}{dt} e^{kt} = y_0 \cdot e^{kt} \cdot k = k y_0 e^{kt}.$$ 5. **Final answer:** $$\boxed{\frac{dy}{dt} = k y_0 e^{kt}}.$$