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
i) Step 1: Apply the product rule. Let u = 3x^2 + 2 and v = 5x - 1. Then (du)/(dx) = 6x and (dv)/(dx) = 5. The product rule states (d)/(dx)(uv) = u(dv)/(dx) + v(du)/(dx). f'(x) = (3x^2 + 2)(5) + (5x - 1)(6x) Step 2: Expand and simplify the expression. f'(x) = 15x^2 + 10 + 30x^2 - 6x f'(x) = 45x^2 - 6x + 10 The derivative is 45x^2 - 6x + 10. ii) Step 1: Apply the quotient rule. Let u = 4x^2 - 3 and v = 2x + 1. Then (du)/(dx) = 8x and (dv)/(dx) = 2. The quotient rule states (dy)/(dx) = (vdu)/(dx) - u(dv)/(dx)v^2. (dy)/(dx) = ((2x+1)(8x) - (4x^2-3)(2))/((2x+1)^2) Step 2: Expand the numerator. (dy)/(dx) = (16x^2 + 8x - (8x^2 - 6))/((2x+1)^2) Step 3: Simplify the numerator. (dy)/(dx) = (16x^2 + 8x - 8x^2 + 6)/((2x+1)^2) (dy)/(dx) = (8x^2 + 8x + 6)/((2x+1)^2) The derivative is (8x^2 + 8x + 6)/((2x+1)^2). iii) Step 1: Simplify the expression using logarithm properties. Recall that a x = x^a. y = e^2 x = e^ x^2 Step 2: Apply the property e^ A = A. y = x^2 Step 3: Differentiate y with respect to x. (dy)/(dx) = (d)/(dx)(x^2) = 2x The derivative is 2x. Got more? Send 'em.