1. **Problem:** Use the limit definition of the derivative to find $f'(x)$ if $f(x) = 3x + 1$. 2. **Limit definition of derivative:** $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. **Apply the function:** $$f(x+h) = 3(x+h) + 1 = 3x + 3h + 1$$ 4. **Substitute into the limit:** $$f'(x) = \lim_{h \to 0} \frac{(3x + 3h + 1) - (3x + 1)}{h} = \lim_{h \to 0} \frac{3h}{h}$$ 5. **Simplify the fraction:** $$\frac{3h}{h} = 3$$ 6. **Evaluate the limit:** $$\lim_{h \to 0} 3 = 3$$ 7. **Final answer:** $$f'(x) = 3$$ This means the derivative of $f(x) = 3x + 1$ is the constant function 3, which matches the slope of the linear function.