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
Here's the solution for question 8. 8. Find (dy)/(dx) for the relation (xy) + (x+y) = 1. Step 1: Differentiate both sides of the equation with respect to x. (d)/(dx)((xy) + (x+y)) = (d)/(dx)(1) Step 2: Apply the chain rule and product rule to differentiate each term. For (xy): Using the chain rule, (d)/(dx)( u) = u · (du)/(dx). Here u = xy. Using the product rule, (d)/(dx)(xy) = 1 · y + x · (dy)/(dx) = y + x(dy)/(dx). So, (d)/(dx)((xy)) = (xy)(y + x(dy)/(dx)). For (x+y): Using the chain rule, (d)/(dx)( v) = - v · (dv)/(dx). Here v = x+y. (d)/(dx)(x+y) = 1 + (dy)/(dx). So, (d)/(dx)((x+y)) = -(x+y)(1 + (dy)/(dx)). For the right side: (d)/(dx)(1) = 0. Step 3: Substitute these derivatives back into the equation. (xy)(y + x(dy)/(dx)) - (x+y)(1 + (dy)/(dx)) = 0 Step 4: Expand the equation. y(xy) + x(xy)(dy)/(dx) - (x+y) - (x+y)(dy)/(dx) = 0 Step 5: Group terms containing (dy)/(dx) on one side and other terms on the other side. x(xy)(dy)/(dx) - (x+y)(dy)/(dx) = (x+y) - y(xy) Step 6: Factor out (dy)/(dx). (dy)/(dx)(x(xy) - (x+y)) = (x+y) - y(xy) Step 7: Solve for (dy)/(dx). (dy)/(dx) = ((x+y) - y(xy))/(x(xy) - (x+y)) The final answer is ((x+y) - y(xy))/(x(xy) - (x+y)). Last free one today — make it count tomorrow, or type /upgrade for unlimited.