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
Step 1: Identify u and v for the quotient rule. Given y = (x^2 + 3)/(x^3 + 4). Let u = x^2 + 3 and v = x^3 + 4. Step 2: Find the derivatives of u and v with respect to x. (du)/(dx) = (d)/(dx)(x^2 + 3) = 2x (dv)/(dx) = (d)/(dx)(x^3 + 4) = 3x^2 Step 3: Apply the quotient rule. The quotient rule states that if y = (u)/(v), then (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2. Substitute the expressions for u, v, (du)/(dx), (dv)/(dx): (dy)/(dx) = ((x^3 + 4)(2x) - (x^2 + 3)(3x^2))/((x^3 + 4)^2) Step 4: Simplify the expression for (dy)/(dx). Expand the numerator: (dy)/(dx) = (2x^4 + 8x - (3x^4 + 9x^2))/((x^3 + 4)^2) (dy)/(dx) = (2x^4 + 8x - 3x^4 - 9x^2)/((x^3 + 4)^2) Combine like terms in the numerator: (dy)/(dx) = (-x^4 - 9x^2 + 8x)/((x^3 + 4)^2) Step 5: Evaluate (dy)/(dx) at x=1. Substitute x=1 into the derivative expression: (dy)/(dx)|_x=1 = (-(1)^4 - 9(1)^2 + 8(1))/(((1)^3 + 4)^2) = (-1 - 9 + 8)/((1 + 4)^2) = (-10 + 8)/((5)^2) = (-2)/(25) The value of (dy)/(dx) at x=1 is: (-2)/(25)