Problem

3 If $f(x)=(2 x+3)(3 x-1)^{5}$, evaluate $f^{\prime}(1)$

Answer

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Answer

\(\boxed{f'(1) = 1264}\)

Steps

Step 1 ：Given the function \(f(x) = (2x + 3)(3x - 1)^5\), we need to find the derivative \(f'(x)\) and evaluate it at \(x = 1\).

Step 2 ：Using the product rule and chain rule, we have \(f'(x) = (2)(3x - 1)^5 + (2x + 3)(5)(3x - 1)^4(3)\).

Step 3 ：Substitute \(x = 1\) into the derivative: \(f'(1) = (2)(3(1) - 1)^5 + (2(1) + 3)(5)(3(1) - 1)^4(3)\).

Step 4 ：Simplify the expression: \(f'(1) = 2(2)^5 + (5)(5)(2)^4(3)\).

Step 5 ：Calculate the final answer: \(f'(1) = 2(32) + 5(5)(16)(3)\).

Step 6 ：\(f'(1) = 64 + 1200\).

Step 7 ：\(\boxed{f'(1) = 1264}\)

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