Question B.6: Find the derivative of y(x) = x³ / (x + 1)² with respect to ...

We can rewrite this function as y(x)= x^{3}(x+1)^{-2} and apply Equation B.30:

\begin{aligned}\frac{d y}{d x} &=(x+1)^{-2} \frac{d}{d x}\left(x^{3}\right)+x^{3} \frac{d}{d x}(x+1)^{-2} \\&=(x+1)^{-2} 3 x^{2}+x^{3}(-2)(x+1)^{-3} \\\frac{d y}{d x} &=\frac{3 x^{2}}{(x+1)^{2}}-\frac{2 x^{3}}{(x+1)^{3}}\end{aligned}

\frac{d}{d x} f(x)=\frac{d}{d x}[g(x) h(x)]=g \frac{d h}{d x}+h \frac{d g}{d x}                   [B.30]