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 differentiate y = (x^2 + 1)^3, we will use the chain rule. Step 1: Identify the outer and inner functions. Let the outer function be f(u) = u^3 and the inner function be u = g(x) = x^2 + 1. Step 2: Differentiate the outer function with respect to u. (df)/(du) = (d)/(du)(u^3) = 3u^2 Step 3: Differentiate the inner function with respect to x. (du)/(dx) = (d)/(dx)(x^2 + 1) = 2x Step 4: Apply the chain rule, which states (dy)/(dx) = (df)/(du) · (du)/(dx). Substitute the expressions from Step 2 and Step 3: (dy)/(dx) = (3u^2)(2x) Step 5: Substitute back u = x^2 + 1 into the expression. (dy)/(dx) = 3(x^2 + 1)^2 (2x) Step 6: Simplify the expression. (dy)/(dx) = 6x(x^2 + 1)^2 The final answer is 6x(x^2 + 1)^2. What's next?