1. **State the problem:** We want to find the derivative of the function $$f(x) = x^2 + 1$$ at the point $$x=2$$. 2. **Recall the definition of the derivative:** The derivative at a point $$x=a$$ is the limit of the average rate of change as $$x$$ approaches $$a$$: $$ f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} $$ This represents the instantaneous rate of change or the slope of the tangent line at $$x=a$$. 3. **Use the table values:** The table shows average rates of change for intervals approaching $$x=2$$ from both sides: - At $$x=1.9$$, average rate is $$3.9$$ - At $$x=1.99$$, average rate is $$3.99$$ - At $$x=1.999$$, average rate is $$3.999$$ - At $$x=2.001$$, average rate is $$4.001$$ - At $$x=2.01$$, average rate is $$4.01$$ - At $$x=2.1$$, average rate is $$4.1$$ 4. **Interpret the limit:** As $$x$$ gets closer to $$2$$, the average rate of change approaches $$4$$. 5. **Confirm by direct differentiation:** Using the power rule, $$ f'(x) = 2x $$ So at $$x=2$$, $$ f'(2) = 2 \times 2 = 4 $$ **Final answer:** The derivative of $$f(x) = x^2 + 1$$ at $$x=2$$ appears to be $$4$$.