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
a) To differentiate Y = 4x^3 + 5x^2 - 9x + 4 with respect to x: Step 1: Apply the power rule for differentiation, (d)/(dx)(ax^n) = anx^n-1, to each term. The derivative of a constant is 0. (dY)/(dx) = (d)/(dx)(4x^3) + (d)/(dx)(5x^2) - (d)/(dx)(9x) + (d)/(dx)(4) Step 2: Differentiate each term. (dY)/(dx) = (4 · 3x^3-1) + (5 · 2x^2-1) - (9 · 1x^1-1) + 0 (dY)/(dx) = 12x^2 + 10x^1 - 9x^0 Step 3: Simplify the expression. (dY)/(dx) = 12x^2 + 10x - 9 The derivative is 12x^2 + 10x - 9. b) To differentiate y = (3x+5)/(6x-9) with respect to x: Step 1: Use the quotient rule, which states that if y = (u)/(v), then (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2. Let u = 3x+5 and v = 6x-9. Step 2: Find the derivatives of u and v. (du)/(dx) = (d)/(dx)(3x+5) = 3 (dv)/(dx) = (d)/(dx)(6x-9) = 6 Step 3: Substitute u, v, (du)/(dx), (dv)/(dx) into the quotient rule formula. (dy)/(dx) = ((6x-9)(3) - (3x+5)(6))/((6x-9)^2) Step 4: Expand and simplify the numerator. (dy)/(dx) = ((18x - 27) - (18x + 30))/((6x-9)^2) (dy)/(dx) = (18x - 27 - 18x - 30)/((6x-9)^2) (dy)/(dx) = (-57)/((6x-9)^2) The derivative is (-57)/((6x-9)^2). That's 2 down. 3 left today — send the next one.