\( y ^ { \prime } = ( \frac { x - 1 } { x ^ { 2 } } ) ^ { \prime } \)

It seems that the calculation for the derivative of \(x^2\) is not supported. Let's calculate it manually. The derivative of \(x^2\) with respect to \(x\) is \(2x\). Now, we can apply the quotient rule to find the derivative of \((x-1)/(x^2)\). According to the quotient rule: \[ \frac{d}{dx}\left(\frac{x-1}{x^2}\right) = \frac{(1)(x^2) - (x-1)(2x)}{(x^2)^2} \] Simplifying this expression will give us the derivative of \((x-1)/(x^2)\). Simplify the expression by following steps: - step0: Solution: \(\frac{\left(1\times x^{2}-\left(x-1\right)\times 2x\right)}{\left(x^{2}\right)^{2}}\) - step1: Remove the parentheses: \(\frac{1\times x^{2}-\left(x-1\right)\times 2x}{\left(x^{2}\right)^{2}}\) - step2: Multiply the exponents: \(\frac{1\times x^{2}-\left(x-1\right)\times 2x}{x^{2\times 2}}\) - step3: Multiply: \(\frac{x^{2}-\left(x-1\right)\times 2x}{x^{2\times 2}}\) - step4: Multiply the terms: \(\frac{x^{2}+2x\left(-x+1\right)}{x^{2\times 2}}\) - step5: Multiply the numbers: \(\frac{x^{2}+2x\left(-x+1\right)}{x^{4}}\) - step6: Add the terms: \(\frac{-x^{2}+2x}{x^{4}}\) - step7: Factor the expression: \(\frac{x\left(-x+2\right)}{x^{4}}\) - step8: Reduce the fraction: \(\frac{-x+2}{x^{3}}\) The derivative of \((x-1)/(x^2)\) is \(\frac{-x+2}{x^3}\). Therefore, \(y' = \frac{-x+2}{x^3}\).