Derivative At 3 Bfdf6D

1. **State the problem:** We need to find the derivative of the function $$f(x) = x^3 - 2$$ at $$x = 3$$ using the alternative form of the derivative. 2. **Recall the alternative form of the derivative:** $$ f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} $$ where $$a = 3$$ in this problem. 3. **Calculate $$f(3)$$:** $$ f(3) = 3^3 - 2 = 27 - 2 = 25 $$ 4. **Calculate $$f(3+h)$$:** $$ f(3+h) = (3+h)^3 - 2 $$ Expand using binomial theorem: $$ (3+h)^3 = 27 + 27h + 9h^2 + h^3 $$ So, $$ f(3+h) = 27 + 27h + 9h^2 + h^3 - 2 = 25 + 27h + 9h^2 + h^3 $$ 5. **Form the difference quotient:** $$ \frac{f(3+h) - f(3)}{h} = \frac{(25 + 27h + 9h^2 + h^3) - 25}{h} = \frac{27h + 9h^2 + h^3}{h} $$ 6. **Simplify the fraction by canceling $$h$$:** $$ \frac{\cancel{h}(27 + 9h + h^2)}{\cancel{h}} = 27 + 9h + h^2 $$ 7. **Take the limit as $$h \to 0$$:** $$ f'(3) = \lim_{h \to 0} (27 + 9h + h^2) = 27 $$ 8. **Conclusion:** The derivative of $$f(x)$$ at $$x=3$$ is $$27$$. **Final answer:** Option C: $$f'(3) = 27$$