1. **Stating the problem:** Find the derivative of a function $f(x)$ with respect to $x$. 2. **Formula used:** The derivative of a function $f(x)$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ This formula gives the instantaneous rate of change of the function at any point $x$. 3. **Important rules:** - The derivative of a constant is 0. - The power rule: $\frac{d}{dx} x^n = n x^{n-1}$. - The sum rule: $\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$. - The product rule: $\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$. - The quotient rule: $\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2}$. - The chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$. 4. **Intermediate work example:** Suppose $f(x) = x^3 + 2x^2 - 5x + 7$. Using the power rule and sum rule: $$f'(x) = \frac{d}{dx} x^3 + \frac{d}{dx} 2x^2 - \frac{d}{dx} 5x + \frac{d}{dx} 7$$ $$= 3x^2 + 2 \cdot 2x^{2-1} - 5 + 0$$ $$= 3x^2 + 4x - 5$$ 5. **Explanation:** We differentiate each term separately using the power rule. Constants vanish because their rate of change is zero. Coefficients multiply the derivative of the variable part. **Final answer:** $$f'(x) = 3x^2 + 4x - 5$$