compute $\frac{d}{dx}f(g(x))$, where $f(x)=\frac{5}{x}$ and $g(x)=4 - x^{2}$\nthe answer is
compute $\frac{d}{dx}f(g(x))$, where $f(x)=\frac{5}{x}$ and $g(x)=4 - x^{2}$\nthe answer is
Answer
Explanation:
Step1: Identify the composite - function
We have (f(x)=\frac{5}{x}) and (g(x)=4 - x^{2}), so (f(g(x))=\frac{5}{4 - x^{2}}).
Step2: Apply 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), (u^\prime=0), (v = 4 - x^{2}), and (v^\prime=-2x). [ \begin{align*} \frac{d}{dx}\left(\frac{5}{4 - x^{2}}\right)&=\frac{0\times(4 - x^{2})-5\times(-2x)}{(4 - x^{2})^{2}}\ &=\frac{10x}{(4 - x^{2})^{2}} \end{align*} ]
Answer:
(\frac{10x}{(4 - x^{2})^{2}})