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
Hey Benoni, good to see you again. Here are the derivatives for the given functions: i) To find the derivative of y = (2+3x^2-x)/(sqrt(1+x)) with respect to x, we use the quotient rule (d)/(dx)((u)/(v)) = (u'v - uv')/(v^2). Step 1: Identify u and v and their derivatives. Let u = 2+3x^2-x, so u' = 6x-1. Let v = sqrt(1+x) = (1+x)^1/2, so v' = (1)/(2)(1+x)^-1/2 = (1)/(2sqrt(1+x)). Step 2: Apply the quotient rule. (dy)/(dx) = (6x-1)sqrt(1+x) - (2+3x^2-x)((1)/(2sqrt(1+x)))(sqrt(1+x))^2 Step 3: Simplify the expression. (dy)/(dx) = (2(6x-1)(1+x) - (2+3x^2-x))/(2sqrt(1+x))1+x (dy)/(dx) = (2(6x^2+5x-1) - (2+3x^2-x))/(2sqrt(1+x)(1+x)) (dy)/(dx) = (12x^2+10x-2 - 2-3x^2+x)/(2(1+x)^3/2) (dy)/(dx) = (9x^2+11x-4)/(2(1+x)^3/2) ii) To find the derivative of y = (x^2-2)/(x^3-2) with respect to x, we use the quotient rule. Step 1: Identify u and v and their derivatives. Let u = x^2-2, so u' = 2x. Let v = x^3-2, so v' = 3x^2. Step 2: Apply the quotient rule. (dy)/(dx) = ((2x)(x^3-2) - (x^2-2)(3x^2))/((x^3-2)^2) Step 3: Simplify the expression. (dy)/(dx) = (2x^4-4x - (3x^4-6x^2))/((x^3-2)^2) (dy)/(dx) = (2x^4-4x - 3x^4+6x^2)/((x^3-2)^2) (dy)/(dx) = (-x^4+6x^2-4x)/((x^3-2)^2) iii) To find the derivative of y = (8x^2+4)/(8x^2-4) with respect to x, we use the quotient rule. Step 1: Identify u and v and their derivatives. Let u = 8x^2+4, so u' = 16x. Let v = 8x^2-4, so v' = 16x. Step 2: Apply the quotient rule. (dy)/(dx) = ((16x)(8x^2-4) - (8x^2+4)(16x))/((8x^2-4)^2) Step 3: Simplify the expression. (dy)/(dx) = (16x[(8x^2-4) - (8x^2+4)])/((8x^2-4)^2) (dy)/(dx) = (16x[8x^2-4 - 8x^2-4])/((8x^2-4)^2) (dy)/(dx) = (16x(-8))/((8x^2-4)^2) (dy)/(dx) = (-128x)/([4(2x^2-1)]^2) (dy)/(dx) = (-128x)/(16(2x^2-1)^2) (dy)/(dx) = (-8x)/((2x^2-1)^2) What's next?