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: Simplify the expression for y using trigonometric identities. We are given the function: y = (1)/(1+^2 x) Recall the Pythagorean trigonometric identity: 1 + ^2 x = ^2 x. Substitute this into the expression for y: y = (1)/(^2 x) Since x = (1)/( x), it follows that (1)/( x) = x. Therefore, (1)/(^2 x) = ^2 x. y = ^2 x Step 2: Differentiate y = ^2 x with respect to x using the chain rule. Let u = x. Then y = u^2. The derivative of y with respect to u is (dy)/(du) = 2u. The derivative of u with respect to x is (du)/(dx) = - x. Applying the chain rule, (dy)/(dx) = (dy)/(du) · (du)/(dx): y' = 2u · (- x) Substitute u = x back into the equation: y' = 2( x)(- x) y' = -2 x x Step 3: Simplify the derivative using a double angle identity. Recall the double angle identity for sine: (2x) = 2 x x. Substitute this into the expression for y': y' = -(2x) The final answer is -(2x). What's next?