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, we will use the quotient rule. The function is y = (x^2 + 7x)/(x+2). Let u = x^2 + 7x and v = x+2. Step 1: 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 2: Apply the quotient rule, which states that if y = (u)/(v), then (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2. (dy)/(dx) = ((x+2)(2x+7) - (x^2+7x)(1))/((x+2)^2) Step 3: Expand the numerator. (x+2)(2x+7) - (x^2+7x)(1) = (2x^2 + 7x + 4x + 14) - (x^2 + 7x) = 2x^2 + 11x + 14 - x^2 - 7x Step 4: Combine like terms in the numerator. (2x^2 - x^2) + (11x - 7x) + 14 = x^2 + 4x + 14 Step 5: Write the final expression for (dy)/(dx). (dy)/(dx) = (x^2 + 4x + 14)/((x+2)^2) The derivative is: (dy)/(dx) = (x^2 + 4x + 14)/((x+2)^2) Send me the next one 📸