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
Welcome back Çhisomm🖤🎀 — missed you this week. To differentiate y = ^2 x ( x), we will use the product rule, which states that if y = u · v, then (dy)/(dx) = u (dv)/(dx) + v (du)/(dx). Step 1: Identify u and v. Let u = ^2 x and v = ( x). Step 2: Differentiate u with respect to x. Using the chain rule for u = ( x)^2: (du)/(dx) = 2( x) · (d)/(dx)( x) = 2 x x Step 3: Differentiate v with respect to x. Using the chain rule for v = ( x): (dv)/(dx) = (1)/( x) · (d)/(dx)( x) = (1)/( x) · (- x) = -( x)/( x) = - x Step 4: Apply the product rule. Substitute u, v, (du)/(dx), and (dv)/(dx) into the product rule formula: (dy)/(dx) = u (dv)/(dx) + v (du)/(dx) (dy)/(dx) = (^2 x)(- x) + (( x))(2 x x) Step 5: Simplify the expression. (dy)/(dx) = -^2 x (( x)/( x)) + 2 x x ( x) (dy)/(dx) = -(^3 x)/( x) + 2 x x ( x) We can factor out x: (dy)/(dx) = x ( -(^2 x)/( x) + 2 x ( x) ) The final answer is x ( -(^2 x)/( x) + 2 x ( x) ). Send me the next one 📸