1. **State the problem:** Find the derivative $f'(x)$ and the domain of the function $f(x) = \ln(15x + 6)$. Also describe the graph shape. 2. **Recall the derivative formula for logarithmic functions:** For $f(x) = \ln(g(x))$, the derivative is $$f'(x) = \frac{g'(x)}{g(x)}$$ where $g(x)$ is the inside function. 3. **Identify $g(x)$ and find $g'(x)$:** Here, $g(x) = 15x + 6$, so $$g'(x) = 15$$ 4. **Apply the derivative formula:** $$f'(x) = \frac{15}{15x + 6}$$ 5. **Determine the domain:** The argument of the logarithm must be positive: $$15x + 6 > 0$$ $$15x > -6$$ $$x > -\frac{6}{15} = -\frac{2}{5}$$ So the domain is $\boxed{\left(-\frac{2}{5}, \infty\right)}$. 6. **Graph shape:** The graph is a logarithmic curve shifted horizontally so its vertical asymptote is at $x = -\frac{2}{5}$. It increases slowly and passes through points where $15x + 6 > 0$. **Final answers:** $$f'(x) = \frac{15}{15x + 6}$$ Domain: $\left(-\frac{2}{5}, \infty\right)$