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 the expression (x^2 + 3)(2x^2 - 1) with respect to x, we can first expand the product and then differentiate each term. Step 1: Expand the given expression. Let y = (x^2 + 3)(2x^2 - 1). Multiply the terms: y = x^2(2x^2) + x^2(-1) + 3(2x^2) + 3(-1) y = 2x^4 - x^2 + 6x^2 - 3 Combine like terms: y = 2x^4 + 5x^2 - 3 Step 2: Differentiate y with respect to x. We use the power rule for differentiation, which states that (d)/(dx)(ax^n) = anx^n-1, and the derivative of a constant is 0. (dy)/(dx) = (d)/(dx)(2x^4) + (d)/(dx)(5x^2) - (d)/(dx)(3) Step 3: Apply the power rule to each term. For 2x^4: (d)/(dx)(2x^4) = 2 · 4x^4-1 = 8x^3 For 5x^2: (d)/(dx)(5x^2) = 5 · 2x^2-1 = 10x For -3: (d)/(dx)(-3) = 0 Step 4: Combine the differentiated terms. (dy)/(dx) = 8x^3 + 10x - 0 (dy)/(dx) = 8x^3 + 10x Comparing this result with the given options: A. 8x^3 - 20x B. 8x^3 + 10x C. x^3 - 10x D. 8x^3 - 10x The correct option is B. The final answer is B. 8x^3 + 10x. That's 2 down. 3 left today — send the next one.