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
Here are the first, second, and third derivatives for both functions: a) For the function y = (2x) Step 1: Find the first derivative, (dy)/(dx). Using the chain rule, (d)/(dx)((ax)) = a(ax): (dy)/(dx) = 2(2x) Step 2: Find the second derivative, (d^2y)/(dx^2). Differentiate (dy)/(dx) = 2(2x). Using the chain rule, (d)/(dx)((ax)) = -a(ax): (d^2y)/(dx^2) = 2(-2(2x)) (d^2y)/(dx^2) = -4(2x) Step 3: Find the third derivative, (d^3y)/(dx^3). Differentiate (d^2y)/(dx^2) = -4(2x). Using the chain rule, (d)/(dx)((ax)) = a(ax): (d^3y)/(dx^3) = -4(2(2x)) (d^3y)/(dx^3) = -8(2x) b) For the function y = x^3 + e^2x Step 1: Find the first derivative, (dy)/(dx). Differentiate x^3 using the power rule ((d)/(dx)(x^n) = nx^n-1) and e^2x using the chain rule ((d)/(dx)(e^ax) = ae^ax): (dy)/(dx) = 3x^3-1 + 2e^2x (dy)/(dx) = 3x^2 + 2e^2x Step 2: Find the second derivative, (d^2y)/(dx^2). Differentiate (dy)/(dx) = 3x^2 + 2e^2x: (d^2y)/(dx^2) = 3(2x^2-1) + 2(2e^2x) (d^2y)/(dx^2) = 6x + 4e^2x Step 3: Find the third derivative, (d^3y)/(dx^3). Differentiate (d^2y)/(dx^2) = 6x + 4e^2x: (d^3y)/(dx^3) = 6(1x^1-1) + 4(2e^2x) (d^3y)/(dx^3) = 6 + 8e^2x (d^3y)/(dx^3) = 6 + 8e^2x 3 done, 2 left today. You're making progress.