Question
Express y=[4x-x^3]/[3x^2+2)^2 as a product and then differentiate. Simplify your answer using positive exponents. Help Please! ty
Answers
Reiny
y = (4x - x^3)(3x^2 + 2)^-2
dy/dx = (4x - x^3)(-2)(3x^2+2)^-3(6x) + (3x^2+2)^-2 (4-3x^2)
= (3x^2+2)^-3 [ -12x(4-x^3) + (3x^2+2)(4-3x^2)]
= (3x^2 + 2)^-3 (3x^4 + 6x^2 - 48x + 8)
= (3x^4 + 6x^2 - 48x + 8)/(3x^2 + 2)^3
dy/dx = (4x - x^3)(-2)(3x^2+2)^-3(6x) + (3x^2+2)^-2 (4-3x^2)
= (3x^2+2)^-3 [ -12x(4-x^3) + (3x^2+2)(4-3x^2)]
= (3x^2 + 2)^-3 (3x^4 + 6x^2 - 48x + 8)
= (3x^4 + 6x^2 - 48x + 8)/(3x^2 + 2)^3