1. **Problem Statement:** Find the derivative of a function using differentiation. 2. **Formula:** The derivative of a function $f(x)$ is given by $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ which represents the rate of change of the function at any point $x$. 3. **Rules:** - The derivative of a constant is 0. - The power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ for any real number $n$. - 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. **Example:** Differentiate $f(x) = 3x^2 + 5x - 7$. 5. **Step-by-step:** - Apply the power rule to each term: - Derivative of $3x^2$ is $3 \times 2 x^{2-1} = 6x$. - Derivative of $5x$ is $5$. - Derivative of constant $-7$ is $0$. 6. **Final answer:** $$f'(x) = 6x + 5$$ This derivative tells us the slope of the tangent line to the curve at any point $x$.