Question: Suppose that $f(x) = 3e^x + 10 \ln(x)$. Find $f'(3)$.
1. **State the problem:** We are given the function $f(x) = 3e^x + 10 \ln(x)$ and need to find the derivative evaluated at $x=3$, i.e., $f'(3)$.
2. **Recall the derivative rules:**
- The derivative of $e^x$ is $e^x$.
- The derivative of $\ln(x)$ is $\frac{1}{x}$.
- The derivative of a sum is the sum of the derivatives.
- Constants multiply through the derivative.
3. **Find the derivative $f'(x)$:**
$$
f'(x) = 3 \frac{d}{dx} e^x + 10 \frac{d}{dx} \ln(x) = 3e^x + 10 \cdot \frac{1}{x} = 3e^x + \frac{10}{x}
$$
4. **Evaluate $f'(3)$:**
$$
f'(3) = 3e^3 + \frac{10}{3}
$$
5. **Final answer:**
$$
f'(3) = 3e^3 + \frac{10}{3}
$$
This is the exact value of the derivative at $x=3$.