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
To find (dy)/(dx) for the given function y = (x^2 + 7x)/(x+2), we will use the quotient rule. The quotient rule states that if y = (u)/(v), then (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2. Step 1: Identify u and v. Let u = x^2 + 7x Let v = x+2 Step 2: Find the derivatives of u and v with respect to x. (du)/(dx) = (d)/(dx)(x^2 + 7x) = 2x + 7 (dv)/(dx) = (d)/(dx)(x+2) = 1 Step 3: Apply the quotient rule formula. (dy)/(dx) = ((x+2)(2x+7) - (x^2+7x)(1))/((x+2)^2) Step 4: Expand and simplify the numerator. Numerator = (x · 2x + x · 7 + 2 · 2x + 2 · 7) - (x^2 + 7x) Numerator = (2x^2 + 7x + 4x + 14) - (x^2 + 7x) Numerator = 2x^2 + 11x + 14 - x^2 - 7x Numerator = (2x^2 - x^2) + (11x - 7x) + 14 Numerator = x^2 + 4x + 14 Step 5: Write the final expression for (dy)/(dx). (dy)/(dx) = (x^2 + 4x + 14)/((x+2)^2) That's 2 down. 3 left today — send the next one.