This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.
ScanSolve AI Answer
To find (dy)/(dx), we will use the chain rule. Step 1: Rewrite the function using exponent notation. y = (12)/(sqrt(3x^3 - 9x^2 + 4x - 8)) y = 12(3x^3 - 9x^2 + 4x - 8)^-(1)/(2) Step 2: Apply the chain rule. Let u = 3x^3 - 9x^2 + 4x - 8. Then y = 12u^-(1)/(2). The chain rule states that (dy)/(dx) = (dy)/(du) · (du)/(dx). First, find (dy)/(du): (dy)/(du) = (d)/(du)(12u^-(1)/(2)) = 12 · (-(1)/(2))u^-(1)/(2) - 1 (dy)/(du) = -6u^-(3)/(2) Next, find (du)/(dx): (du)/(dx) = (d)/(dx)(3x^3 - 9x^2 + 4x - 8) (du)/(dx) = 3(3x^3-1) - 9(2x^2-1) + 4(1x^1-1) - 0 (du)/(dx) = 9x^2 - 18x + 4 Step 3: Multiply (dy)/(du) and (du)/(dx) to find (dy)/(dx). (dy)/(dx) = (-6u^-(3)/(2)) · (9x^2 - 18x + 4) Step 4: Substitute u = 3x^3 - 9x^2 + 4x - 8 back into the expression for (dy)/(dx). (dy)/(dx) = -6(3x^3 - 9x^2 + 4x - 8)^-(3)/(2)(9x^2 - 18x + 4) Step 5: Rewrite the expression with positive exponents and in radical form. (dy)/(dx) = (-6(9x^2 - 18x + 4))/((3x^3 - 9x^2 + 4x - 8)^3)2 (dy)/(dx) = (-6(9x^2 - 18x + 4))/((sqrt(3x^3 - 9x^2 + 4x - 8))^3) The final answer is (dy)/(dx) = (-6(9x^2 - 18x + 4))/((sqrt(3x^3 - 9x^2 + 4x - 8))^3).