1. **Problem Statement:** Find the derivative of a function using the rules of derivatives. 2. **Derivative Rules:** The most common rules are: - 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: $$\frac{d}{dx} f(g(x)) = f'(g(x)) g'(x)$$ 3. **Example:** Find the derivative of $$y = 3x^4 - 5x^2 + 6x - 7$$. 4. **Step-by-step:** - Apply the power rule to each term: $$\frac{d}{dx} 3x^4 = 3 \times 4 x^{4-1} = 12x^3$$ $$\frac{d}{dx} (-5x^2) = -5 \times 2 x^{2-1} = -10x$$ $$\frac{d}{dx} 6x = 6$$ $$\frac{d}{dx} (-7) = 0$$ 5. **Combine results:** $$y' = 12x^3 - 10x + 6$$ 6. **Explanation:** We differentiated each term separately using the power rule and constants rule, then summed the results because differentiation is linear. **Final answer:** $$\boxed{y' = 12x^3 - 10x + 6}$$