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: Find the first derivative (dy)/(dx) of y = 8sqrt(x). First, rewrite y using exponent notation: y = 8x^1/2. Now, differentiate using the power rule (d)/(dx)(x^n) = nx^n-1: (dy)/(dx) = 8 · (1)/(2) x^(1)/(2)-1 = 4x^-1/2 Step 2: Find the second derivative (d^2y)/(dx^2). Differentiate (dy)/(dx) = 4x^-1/2 again: (d^2y)/(dx^2) = 4 · (-(1)/(2)) x^-(1)/(2)-1 = -2x^-3/2 Step 3: Substitute y, (dy)/(dx), and (d^2y)/(dx^2) into the left-hand side of the equation 4x^2 (d^2y)/(dx^2) + 4x (dy)/(dx) = y. The left-hand side (LHS) is: 4x^2 (-2x^-3/2) + 4x (4x^-1/2) Step 4: Simplify the expression. Multiply the terms: -8x^2 - 3/2 + 16x^1 - 1/2 Combine the exponents: -8x^4/2 - 3/2 + 16x^2/2 - 1/2 -8x^1/2 + 16x^1/2 Combine the coefficients: (-8+16)x^1/2 = 8x^1/2 Rewrite in radical form: 8sqrt(x) Step 5: Compare the simplified LHS with y. We found that the LHS simplifies to 8sqrt(x). Given that y = 8sqrt(x). Therefore, 4x^2 (d^2y)/(dx^2) + 4x (dy)/(dx) = y is shown. That's 2 down. 3 left today — send the next one.