The Definition of the Derivative — Question 4

Question 4 - Solution

We are given: $$f(x) = x^2$$

Using the definition of the derivative at $x = 3$: $$f'(3) = \lim_{h \to 0} \frac{f(3 + h) - f(3)}{h}$$

Compute each term: $$f(3 + h) = (3 + h)^2 = 9 + 6h + h^2$$ $$f(3) = 3^2 = 9$$

Substitute into the limit: $$f'(3) = \lim_{h \to 0} \frac{(9 + 6h + h^2) - 9}{h} = \lim_{h \to 0} \frac{6h + h^2}{h}$$

Factor $h$ from the numerator: $$= \lim_{h \to 0} \frac{h(6 + h)}{h}$$

Cancel $h$: $$= \lim_{h \to 0} (6 + h)$$

Now take the limit: $$f'(3) = \boxed{6}$$