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 (dx)/(dy) for the given equation x = (3)/(4) (2y - ()/(2)), we will use the chain rule. Step 1: Identify the outer and inner functions. Let the outer function be f(u) = (3)/(4) u and the inner function be u = 2y - ()/(2). Step 2: Differentiate the inner function with respect to y. (du)/(dy) = (d)/(dy)(2y - ()/(2)) = 2 Step 3: Differentiate the outer function with respect to u. Recall that (d)/(du)( u) = -^2 u. (dx)/(du) = (d)/(du)((3)/(4) u) = (3)/(4) (-^2 u) = -(3)/(4) ^2 u Step 4: Apply the chain rule, (dx)/(dy) = (dx)/(du) · (du)/(dy). (dx)/(dy) = (-(3)/(4) ^2 u) · (2) Step 5: Substitute u = 2y - ()/(2) back into the expression and simplify. (dx)/(dy) = -(3)/(4) ^2 (2y - ()/(2)) · 2 (dx)/(dy) = -(3 · 2)/(4) ^2 (2y - ()/(2)) (dx)/(dy) = -(6)/(4) ^2 (2y - ()/(2)) (dx)/(dy) = -(3)/(2) ^2 (2y - ()/(2)) The final answer is -(3)/(2) ^2 (2y - ()/(2)).