1. We need to find the derivative $f'(x)$ of the function $f(x) = 4x + 7$ using the definition of the derivative. 2. The definition of the derivative at a point $x$ is given by: $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ 3. Substitute $f(x) = 4x + 7$ into the formula: $$f'(x) = \lim_{h \to 0} \frac{4(x+h) + 7 - (4x + 7)}{h}$$ 4. Simplify the numerator: $$4(x+h) + 7 - 4x - 7 = 4x + 4h + 7 - 4x -7 = 4h$$ 5. So the expression becomes: $$f'(x) = \lim_{h \to 0} \frac{4h}{h}$$ 6. Simplify the fraction: $$\frac{4h}{h} = 4$$ (for $h \neq 0$) 7. Take the limit as $h$ approaches 0: $$f'(x) = \lim_{h \to 0} 4 = 4$$ 8. Therefore, the derivative of $f(x) = 4x + 7$ is: $$f'(x) = 4$$ This result means that at any point $x$, the slope of the function $4x + 7$ is 4, which aligns with the coefficient of $x$ in the linear function.