1. **State the problem:** Differentiate the function $$y = 11 e^{0.05x} + 4$$ with respect to $$x$$. 2. **Recall the differentiation rule for exponential functions:** The derivative of $$e^{u(x)}$$ with respect to $$x$$ is $$e^{u(x)} \cdot u'(x)$$, where $$u(x)$$ is a function of $$x$$. 3. **Apply the rule:** Here, $$u(x) = 0.05x$$, so $$u'(x) = 0.05$$. 4. **Differentiate each term:** - The derivative of $$11 e^{0.05x}$$ is $$11 \cdot e^{0.05x} \cdot 0.05 = 0.55 e^{0.05x}$$. - The derivative of the constant $$4$$ is $$0$$. 5. **Combine the results:** $$\frac{dy}{dx} = 0.55 e^{0.05x}$$ **Final answer:** $$\boxed{\frac{dy}{dx} = 0.55 e^{0.05x}}$$