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 show that (d)/(dx)( x) = -^2 x using the quotient rule, we start with x = ( x)/( x). Let u = x and v = x. Step 1: Find the derivatives of u and v. (du)/(dx) = (d)/(dx)( x) = - x (dv)/(dx) = (d)/(dx)( x) = x Step 2: Apply the quotient rule, which states that if y = (u)/(v), then (dy)/(dx) = (v du)/(dx) - u (dv)/(dx)v^2. (d)/(dx)( x) = (( x)(- x) - ( x)( x))/(( x)^2) Step 3: Simplify the numerator. (d)/(dx)( x) = (-^2 x - ^2 x)/(^2 x) Step 4: Factor out -1 from the numerator. (d)/(dx)( x) = (-(^2 x + ^2 x))/(^2 x) Step 5: Use the Pythagorean identity ^2 x + ^2 x = 1. (d)/(dx)( x) = (-1)/(^2 x) Step 6: Express the result in terms of x, where x = (1)/( x). (d)/(dx)( x) = -((1)/( x))^2 = -^2 x Thus, we have shown that: (d)/(dx)( x) = -^2 x