Final answer:
The derivative f'(x) of the function f(x) = √(4x² + 4x + 3) is calculated using the chain rule and is (4x + 2) / √(4x² + 4x + 3).
Step-by-step explanation:
The student is asking to find the derivative of the function f(x) = √(4x² + 4x + 3). To find f'(x), we need to apply the chain rule. The chain rule in calculus is a formula to compute the derivative of a composite function. The composite function here is the square root of a quadratic, which means we have to differentiate the outer function (the square root) and multiply it by the derivative of the inner function (the quadratic).
The derivative of the square root function √(u) with respect to u is ⅐ * u'∙ ∙. The derivative of the inside function 4x² + 4x + 3 with respect to x is 8x + 4. Now, we apply the chain rule:
f'(x) = ⅐ * (8x + 4) = ⅐(8x + 4) / (2√(4x² + 4x + 3)) = (4x + 2) / √(4x² + 4x + 3)