\( f ^ { \prime } ( x ) = ( \frac { x ^ { 2 } } { x + 1 } ) ^ { \prime } = \)

To find \( f^{\prime}(x) \), we need to apply the quotient rule here, since we have a function that is a quotient of two functions. The quotient rule states that if you have \( \frac{u}{v} \), then its derivative is given by \( \frac{u'v - uv'}{v^2} \). For \( f(x) = \frac{x^2}{x + 1} \), let \( u = x^2 \) and \( v = x + 1 \). Then \( u' = 2x \) and \( v' = 1 \). Now, applying the quotient rule: \[ f^{\prime}(x) = \frac{(2x)(x + 1) - (x^2)(1)}{(x + 1)^2} = \frac{2x^2 + 2x - x^2}{(x + 1)^2} = \frac{x^2 + 2x}{(x + 1)^2} \] Now, let's simplify that. Moreover, if you ever find yourself calculating derivatives like this, keep in mind that it’s super handy to differentiate each component systematically. Don’t forget to double-check your arithmetic, especially when dealing with polynomials – common mistakes can slip in when simplifying fractions!