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: Find (dx)/(dt) using the quotient rule. Given x = (1-t^2)/(1+t^2). Let u = 1-t^2 and v = 1+t^2. Then (du)/(dt) = -2t and (dv)/(dt) = 2t. The quotient rule is (d)/(dt)((u)/(v)) = (u'v - uv')/(v^2). (dx)/(dt) = ((-2t)(1+t^2) - (1-t^2)(2t))/((1+t^2)^2) (dx)/(dt) = (-2t - 2t^3 - (2t - 2t^3))/((1+t^2)^2) (dx)/(dt) = (-2t - 2t^3 - 2t + 2t^3)/((1+t^2)^2) (dx)/(dt) = (-4t)/((1+t^2)^2) Step 2: Find (dy)/(dt) using the quotient rule. Given y = (2t)/(1+t^2). Let u = 2t and v = 1+t^2. Then (du)/(dt) = 2 and (dv)/(dt) = 2t. (dy)/(dt) = ((2)(1+t^2) - (2t)(2t))/((1+t^2)^2) (dy)/(dt) = (2 + 2t^2 - 4t^2)/((1+t^2)^2) (dy)/(dt) = (2 - 2t^2)/((1+t^2)^2) (dy)/(dt) = (2(1 - t^2))/((1+t^2)^2) Step 3: Calculate (dy)/(dx) using the chain rule (dy)/(dx) = (dy/dt)/(dx/dt). (dy)/(dx) = (2(1 - t^2))/((1+t^2)^2)(-4t)/((1+t^2)^2) (dy)/(dx) = (2(1 - t^2))/((1+t^2)^2) · ((1+t^2)^2)/(-4t) (dy)/(dx) = (2(1 - t^2))/(-4t) (dy)/(dx) = -(1 - t^2)/(2t) Step 4: Calculate (x)/(y). x = (1-t^2)/(1+t^2) y = (2t)/(1+t^2) (x)/(y) = (1-t^2)/(1+t^2)(2t)/(1+t^2) (x)/(y) = (1-t^2)/(1+t^2) · (1+t^2)/(2t) (x)/(y) = (1-t^2)/(2t) Step 5: Substitute (dy)/(dx) and (x)/(y) into the expression (dy)/(dx) + (x)/(y). (dy)/(dx) + (x)/(y) = -(1 - t^2)/(2t) + (1-t^2)/(2t) (dy)/(dx) + (x)/(y) = 0 Thus, it is proven that (dy)/(dx) + (x)/(y) = 0. That's 2 down. 3 left today — send the next one.