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