This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Rewrite the function f(x) to simplify differentiation. f(x) = (x^2 + 2x + 4)/(x^2 + 2x) We can split the fraction: f(x) = (x^2 + 2x)/(x^2 + 2x) + (4)/(x^2 + 2x) f(x) = 1 + (4)/(x^2 + 2x) Now, express the second term with a negative exponent: f(x) = 1 + 4(x^2 + 2x)^-1 Step 2: Differentiate f(x) using the chain rule. The derivative of a constant is 0. For the second term, we use the chain rule (d)/(dx)[c · g(x)^n] = c · n · g(x)^n-1 · g'(x). Here, c=4, n=-1, and g(x) = x^2 + 2x. First, find g'(x): g'(x) = (d)/(dx)(x^2 + 2x) = 2x + 2 Step 3: Apply the differentiation rules. f'(x) = (d)/(dx)(1) + (d)/(dx)(4(x^2 + 2x)^-1) f'(x) = 0 + 4 · (-1)(x^2 + 2x)^-1-1 · (2x + 2) f'(x) = -4(x^2 + 2x)^-2 (2x + 2) Step 4: Simplify the expression. f'(x) = (-4(2x + 2))/((x^2 + 2x)^2) f'(x) = (-8x - 8)/((x^2 + 2x)^2) We can factor out -8 from the numerator: f'(x) = (-8(x + 1))/((x^2 + 2x)^2) The derivative f'(x) is (-8(x + 1))/((x^2 + 2x)^2). Drop the next question.