differentiate with respect to the independent variable.\n\n$f(x)=\\frac{5 - 2x^{3}}{1 + x}$\n\n$f(x)=\\square$
differentiate with respect to the independent variable.\n\n$f(x)=\\frac{5 - 2x^{3}}{1 + x}$\n\n$f(x)=\\square$
Answer
Explanation:
Step1: Identify the quotient rule
The quotient rule states that if (y = \frac{u}{v}), then (y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}). Here, (u = 5-2x^{3}), (u^\prime=-6x^{2}), (v = 1 + x), (v^\prime=1).
Step2: Apply the quotient rule
[ \begin{align*} f^\prime(x)&=\frac{(-6x^{2})(1 + x)-(5-2x^{3})(1)}{(1 + x)^{2}}\ &=\frac{-6x^{2}-6x^{3}-5 + 2x^{3}}{(1 + x)^{2}}\ &=\frac{-4x^{3}-6x^{2}-5}{(1 + x)^{2}} \end{align*} ]
Answer:
(\frac{-4x^{3}-6x^{2}-5}{(1 + x)^{2}})