1. Problem statement: Find the derivative of $f(x) = (3x + 1)^2$ using three methods: the chain rule, the product rule, and by expanding the square. Then compare the computational effort. 2. Method 1: Chain rule - Formula: If $f(x) = (g(x))^2$, then $f'(x) = 2g(x) \cdot g'(x)$. - Here, $g(x) = 3x + 1$, so $g'(x) = 3$. - Therefore, $$f'(x) = 2(3x + 1) \cdot 3 = 6(3x + 1) = 18x + 6.$$ 3. Method 2: Product rule - Rewrite $f(x) = (3x + 1)(3x + 1)$. - Product rule: $(uv)' = u'v + uv'$. - Let $u = 3x + 1$, $v = 3x + 1$, so $u' = 3$, $v' = 3$. - Then, $$f'(x) = 3(3x + 1) + (3x + 1)3 = 3(3x + 1) + 3(3x + 1) = 6(3x + 1) = 18x + 6.$$ 4. Method 3: Expand and differentiate - Expand: $f(x) = (3x + 1)^2 = 9x^2 + 6x + 1$. - Differentiate term-by-term: $$f'(x) = 18x + 6.$$ 5. Comparison of effort - Chain rule: straightforward, requires identifying inner function and its derivative. - Product rule: more steps, applying product rule twice. - Expansion: requires algebraic expansion first, then simple differentiation. Final answer: $$f'(x) = 18x + 6.$$