Final answer:
To find the derivative of the given function (x-1)(3x+4), we can use the product rule. The derivative of the product is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function. Expanding the product first and then differentiating also leads to the same result.
Step-by-step explanation:
(a) To find the derivative of the given function f(x) = (x-1)(3x+4) using the product rule, we apply the following formula: d/dx [f(x)g(x)] = f'(x)g(x) + f(x)g'(x). Let's denote f(x) = (x-1) and g(x) = (3x+4). Then, we have f'(x) = 1 and g'(x) = 3. Applying the product rule, we get:
f'(x)g(x) + f(x)g'(x) = (1)(3x+4) + (x-1)(3) = 3x + 4 + 3x - 3 = 6x + 1
(b) To find the derivative by expanding the product first, we multiply out the two factors:
f(x) = (x-1)(3x+4) = 3x^2 + 4x - 3x - 4 = 3x^2 + x - 4
Now, we differentiate the expanded form of f(x) to find the derivative:
f'(x) = 6x + 1, which matches the result from part (a) and verifies that our answer is correct.