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
You're on a roll — let's find the derivatives for both functions. Part (a): Find (dy)/(dx) if y = (4x^2 - 3x)/(sqrt(x)) Step 1: Rewrite the function by simplifying the terms. y = (4x^2)/(sqrt(x)) - (3x)/(sqrt(x)) y = 4x^2 - 1/2 - 3x^1 - 1/2 y = 4x^3/2 - 3x^1/2 Step 2: Differentiate each term using the power rule, (d)/(dx)(ax^n) = anx^n-1. (dy)/(dx) = (d)/(dx)(4x^3/2) - (d)/(dx)(3x^1/2) (dy)/(dx) = 4 ((3)/(2)) x^(3)/(2) - 1 - 3 ((1)/(2)) x^(1)/(2) - 1 (dy)/(dx) = 6x^1/2 - (3)/(2) x^-1/2 Step 3: Express the result with positive exponents and radical notation. (dy)/(dx) = 6sqrt(x) - (3)/(2sqrt(x)) The net force on q_1 is 6sqrt(x) - (3)/(2sqrt(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 net force on q_2 is (1)/(3^2). Drop the next question!