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
To find the first derivative (dy)/(dx) of the function y = (2x^3+4)/((5x-6)^2), we use the quotient rule. Step 1: Identify u and v and their derivatives. Let u = 2x^3+4. Let v = (5x-6)^2. Find the derivative of u: u' = (d)/(dx)(2x^3+4) = 6x^2 Find the derivative of v using the chain rule: v' = (d)/(dx)((5x-6)^2) Let w = 5x-6, so v = w^2. Then (dv)/(dw) = 2w and (dw)/(dx) = 5. v' = (dv)/(dw) · (dw)/(dx) = 2(5x-6) · 5 = 10(5x-6) Step 2: Apply the quotient rule. The quotient rule states that if y = (u)/(v), then (dy)/(dx) = (u'v - uv')/(v^2). Substitute u, v, u', v' into the formula: (dy)/(dx) = ((6x^2)(5x-6)^2 - (2x^3+4)(10(5x-6)))/(((5x-6)^2)^2) (dy)/(dx) = (6x^2(5x-6)^2 - 10(2x^3+4)(5x-6))/((5x-6)^4) Step 3: Simplify the expression. Factor out (5x-6) from the numerator: (dy)/(dx) = ((5x-6)[6x^2(5x-6) - 10(2x^3+4)])/((5x-6)^4) Cancel one (5x-6) term from the numerator and denominator: (dy)/(dx) = (6x^2(5x-6) - 10(2x^3+4))/((5x-6)^3) Expand the terms in the numerator: (dy)/(dx) = (30x^3 - 36x^2 - 20x^3 - 40)/((5x-6)^3) Combine like terms in the numerator: (dy)/(dx) = ((30x^3 - 20x^3) - 36x^2 - 40)/((5x-6)^3) (dy)/(dx) = (10x^3 - 36x^2 - 40)/((5x-6)^3) That's 2 down. 3 left today — send the next one.