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
My apologies for the oversight in the previous response. You are absolutely right to ask for a clear check. I misread the denominator in part (a). Let's re-solve both parts carefully. Part (a): Find (dy)/(dx) if y = (4x^2 - 3x)/(8sqrt(x)) Step 1: Rewrite the function by separating the terms and simplifying using exponent rules. y = (4x^2)/(8sqrt(x)) - (3x)/(8sqrt(x)) y = (4x^2)/(8x^1/2) - (3x)/(8x^1/2) y = (1)/(2)x^2 - 1/2 - (3)/(8)x^1 - 1/2 y = (1)/(2)x^3/2 - (3)/(8)x^1/2 Step 2: Differentiate each term using the power rule, (d)/(dx)(ax^n) = anx^n-1. (dy)/(dx) = (d)/(dx)((1)/(2)x^3/2) - (d)/(dx)((3)/(8)x^1/2) (dy)/(dx) = (1)/(2) ((3)/(2)) x^(3)/(2) - 1 - (3)/(8) ((1)/(2)) x^(1)/(2) - 1 (dy)/(dx) = (3)/(4)x^1/2 - (3)/(16) x^-1/2 Step 3: Express the result with positive exponents and radical notation. (dy)/(dx) = (3)/(4)sqrt(x) - (3)/(16sqrt(x)) The derivative for part (a) is (3)/(4)sqrt(x) - (3)/(16sqrt(x)). Part (b): Find (dy)/(dx) if y = [3]3x+4 Step 1: Rewrite the function using fractional exponents. y = (3x+4)^1/3 Step 2: Differentiate using the chain rule, (d)/(dx)(u^n) = nu^n-1 (du)/(dx). Here, u = 3x+4 and n = (1)/(3). (dy)/(dx) = (1)/(3) (3x+4)^(1)/(3) - 1 · (d)/(dx)(3x+4) (dy)/(dx) = (1)/(3) (3x+4)^-2/3 · 3 Step 3: Simplify the expression. (dy)/(dx) = (3x+4)^-2/3 Step 4: Express the result with positive exponents and radical notation. (dy)/(dx) = (1)/((3x+4)^2/3) (dy)/(dx) = (1)/(3^2) The derivative for part (b) is (1)/(3^2). Last free one today — make it count tomorrow, or type /upgrade for unlimited.