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
Welcome back Lotum — missed you this week. Here are the solutions to the problems: 10. (i) a) Differentiate ((3x^3)/( x)) with respect to x. Step 1: Use logarithm properties to simplify the expression. ((3x^3)/( x)) = (3x^3) - ( x) = 3 + (x^3) - ( x) = 3 + 3 x - ( x) Step 2: Differentiate each term with respect to x. (d)/(dx)( 3 + 3 x - ( x)) = (d)/(dx)( 3) + (d)/(dx)(3 x) - (d)/(dx)(( x)) = 0 + 3((1)/(x)) - ((1)/( x) · x) = (3)/(x) - ( x)/( x) = (3)/(x) - x The derivative is (3)/(x) - x. 10. (i) b) Find the equation of the tangent to the curve defined parametrically by x = t^3, y = t(1 - (1)/(4)t^3) at the point where t = -1. Step 1: Find the coordinates (x, y) of the point at t = -1. x = (-1)^3 = -1 y = (-1)(1 - (1)/(4)(-1)^3) = -1(1 - (1)/(4)(-1)) = -1(1 + (1)/(4)) = -1((5)/(4)) = -(5)/(4) The point is (-1, -(5)/(4)). Step 2: Find (dx)/(dt) and (dy)/(dt). x = t^3 (dx)/(dt) = 3t^2 y = t - (1)/(4)t^4 (dy)/(dt) = 1 - (1)/(4)(4t^3) = 1 - t^3 Step 3: Find the gradient $dydx