1. **Problem statement:** Find the derivative of the function $$y = \log_5(7x) + 8^{9x}$$. 2. **Recall the formulas and rules:** - The derivative of $$\log_a(u)$$ with respect to $$x$$ is $$\frac{1}{u \ln(a)} \cdot \frac{du}{dx}$$, where $$a$$ is the base of the logarithm. - The derivative of an exponential function $$a^{v}$$ with respect to $$x$$ is $$a^{v} \ln(a) \cdot \frac{dv}{dx}$$. 3. **Apply the derivative to each term:** - For $$\log_5(7x)$$: - Let $$u = 7x$$, so $$\frac{du}{dx} = 7$$. - Derivative is $$\frac{1}{7x \ln(5)} \cdot 7 = \frac{7}{7x \ln(5)} = \frac{1}{x \ln(5)}$$. - For $$8^{9x}$$: - Let $$v = 9x$$, so $$\frac{dv}{dx} = 9$$. - Derivative is $$8^{9x} \ln(8) \cdot 9 = 9 \ln(8) \cdot 8^{9x}$$. 4. **Combine the derivatives:** $$\frac{dy}{dx} = \frac{1}{x \ln(5)} + 9 \ln(8) \cdot 8^{9x}$$. 5. **Final answer:** $$\boxed{\frac{dy}{dx} = \frac{1}{x \ln(5)} + 9 \ln(8) \cdot 8^{9x}}$$