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
The problem asks to differentiate y = (3x^2 + 5)(4x^4 + 3) using the product rule. The product rule states that if y = UV, then (dy)/(dx) = U(dV)/(dx) + V(dU)/(dx). Step 1: Identify U and V and calculate their derivatives. Let U = 3x^2 + 5. Then (dU)/(dx) = (d)/(dx)(3x^2 + 5) = 3(2x^2-1) + 0 = 6x. This matches your calculation for (dU)/(dx). Let V = 4x^4 + 3. Then (dV)/(dx) = (d)/(dx)(4x^4 + 3) = 4(4x^4-1) + 0 = 16x^3. Correction: Your calculation for (dV)/(dx) was 16x, but it should be 16x^3. Step 2: Apply the product rule formula. Using the correct derivatives: (dy)/(dx) = (3x^2 + 5)(16x^3) + (4x^4 + 3)(6x) Note: In your written work, you used 16x for (dV)/(dx) and 4x^3+3 for V in this step. However, your subsequent expansion suggests you implicitly used the correct terms. Step 3: Expand the terms. (dy)/(dx) = (3x^2)(16x^3) + (5)(16x^3) + (4x^4)(6x) + (3)(6x) (dy)/(dx) = 48x^2+3 + 80x^3 + 24x^4+1 + 18x (dy)/(dx) = 48x^5 + 80x^3 + 24x^5 + 18x This expansion matches your work. Step 4: Combine like terms. (dy)/(dx) = (48x^5 + 24x^5) + 80x^3 + 18x (dy)/(dx) = 72x^5 + 80x^3 + 18x This matches your final answer. Despite the minor errors in writing down (dV)/(dx) and V in the intermediate steps, your final result is correct because you applied the correct terms during the expansion. The final answer is 72x^5 + 80x^3 + 18x. That's 2 down. 3 left today — send the next one.