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
Step 1: Differentiate y = 3x^6. To differentiate y = 3x^6, use the power rule for differentiation, which states that if y = ax^n, then (dy)/(dx) = n · ax^n-1. Here, a = 3 and n = 6. (dy)/(dx) = 6 · 3x^6-1 (dy)/(dx) = 18x^5 Step 2: Find (dy)/(dx) of y = 2x^3 - 3x^2 + 12x. To differentiate y = 2x^3 - 3x^2 + 12x, differentiate each term separately using the power rule. Differentiate the first term, 2x^3: (d)/(dx)(2x^3) = 3 · 2x^3-1 = 6x^2 Differentiate the second term, -3x^2: (d)/(dx)(-3x^2) = 2 · (-3)x^2-1 = -6x Differentiate the third term, 12x: (d)/(dx)(12x) = 1 · 12x^1-1 = 12x^0 = 12 · 1 = 12 Combine the derivatives of each term: (dy)/(dx) = 6x^2 - 6x + 12 a) The derivative of y = 3x^6 is 18x^5. b) The derivative (dy)/(dx) of y = 2x^3 - 3x^2 + 12x is 6x^2 - 6x + 12.