Derivative Method 4Dcf80

1. The problem is to understand and apply the derivative method in calculus. 2. The derivative of a function $f(x)$ is defined as $$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$ which represents the rate of change or slope of the function at any point $x$. 3. Important rules for derivatives include: - Power rule: $$\frac{d}{dx} x^n = n x^{n-1}$$ - Sum rule: $$\frac{d}{dx} [f(x) + g(x)] = f'(x) + g'(x)$$ - Product rule: $$\frac{d}{dx} [f(x) g(x)] = f'(x) g(x) + f(x) g'(x)$$ - Quotient rule: $$\frac{d}{dx} \left( \frac{f(x)}{g(x)} \right) = \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2}$$ - Chain rule: $$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$$ 4. To find the derivative of a function, apply these rules step-by-step, simplifying and canceling common factors where possible. 5. For example, to find the derivative of $$f(x) = 3x^2 + 5x - 4$$: - Apply the power rule to each term: $$\frac{d}{dx} 3x^2 = 3 \cdot 2 x^{2-1} = 6x$$ $$\frac{d}{dx} 5x = 5$$ $$\frac{d}{dx} (-4) = 0$$ - Combine results: $$f'(x) = 6x + 5$$ 6. This derivative tells us the slope of the function $f(x)$ at any point $x$. This completes the explanation of the derivative method.