1. **State the problem:** Find the derivative $\frac{dy}{dx}$ of the function $$y = (2 + 6x)^{\frac{5}{x}}$$ and then evaluate the slope and the value of $y$ at $x=1$. 2. **Use logarithmic differentiation:** Since the function is of the form $f(x)^{g(x)}$, take the natural logarithm of both sides: $$\ln y = \frac{5}{x} \ln(2 + 6x)$$ 3. **Differentiate both sides with respect to $x$:** Using the chain rule on the left and product rule on the right: $$\frac{1}{y} \frac{dy}{dx} = \frac{d}{dx} \left( \frac{5}{x} \ln(2 + 6x) \right)$$ 4. **Apply the product rule:** Let $u = \frac{5}{x}$ and $v = \ln(2 + 6x)$, then $$\frac{d}{dx}(uv) = u'v + uv'$$ Calculate derivatives: $$u' = -\frac{5}{x^2}$$ $$v' = \frac{6}{2 + 6x}$$ So, $$\frac{d}{dx} \left( \frac{5}{x} \ln(2 + 6x) \right) = -\frac{5}{x^2} \ln(2 + 6x) + \frac{5}{x} \cdot \frac{6}{2 + 6x}$$ 5. **Substitute back:** $$\frac{1}{y} \frac{dy}{dx} = -\frac{5}{x^2} \ln(2 + 6x) + \frac{30}{x(2 + 6x)}$$ 6. **Multiply both sides by $y$ to isolate $\frac{dy}{dx}$:** $$\frac{dy}{dx} = y \left(-\frac{5}{x^2} \ln(2 + 6x) + \frac{30}{x(2 + 6x)} \right)$$ 7. **Recall $y = (2 + 6x)^{\frac{5}{x}}$ and substitute:** $$\frac{dy}{dx} = (2 + 6x)^{\frac{5}{x}} \left(-\frac{5}{x^2} \ln(2 + 6x) + \frac{30}{x(2 + 6x)} \right)$$ 8. **Evaluate the slope at $x=1$:** Calculate each term: $$\ln(2 + 6 \cdot 1) = \ln(8)$$ $$y(1) = 8^{5} = 32768$$ Calculate derivative terms: $$-\frac{5}{1^2} \ln(8) = -5 \ln(8)$$ $$\frac{30}{1 \cdot 8} = \frac{30}{8} = 3.75$$ So, $$\frac{dy}{dx}\bigg|_{x=1} = 32768 \times (-5 \ln(8) + 3.75)$$ 9. **Calculate numerical value:** $$\ln(8) = \ln(2^3) = 3 \ln(2) \approx 3 \times 0.6931 = 2.0794$$ Then, $$-5 \ln(8) + 3.75 = -5 \times 2.0794 + 3.75 = -10.397 + 3.75 = -6.647$$ Finally, $$\frac{dy}{dx}\bigg|_{x=1} = 32768 \times (-6.647) \approx -217,800$$ 10. **Value of $y$ at $x=1$:** $$y(1) = 8^{5} = 32768$$ **Final answers:** $$\frac{dy}{dx} = (2 + 6x)^{\frac{5}{x}} \left(-\frac{5}{x^2} \ln(2 + 6x) + \frac{30}{x(2 + 6x)} \right)$$ Slope at $x=1$ is approximately $$-217,800$$ Value of $y$ at $x=1$ is $$32768$$