exercsies 3.3 differentiation rules\nscore: 27/40 answered: 13/19\nquestion 14\ntextbook videos +\nif (…

exercsies 3.3 differentiation rules\nscore: 27/40 answered: 13/19\nquestion 14\ntextbook videos +\nif ( f(x)=\frac{3 - x^{2}}{7 + x^{2}} ), find:\n( f^{prime}(x)= )\nquestion help: video message instructor\nsubmit question jump to answer

exercsies 3.3 differentiation rules\nscore: 27/40 answered: 13/19\nquestion 14\ntextbook videos +\nif ( f(x)=\frac{3 - x^{2}}{7 + x^{2}} ), find:\n( f^{prime}(x)= )\nquestion help: video message instructor\nsubmit question jump to answer

Answer

Explanation:

Step1: Identify the quotient rule

The quotient rule states that if (y = \frac{u}{v}), then (y'=\frac{u'v - uv'}{v^{2}}). Here, (u = 3 - x^{2}), (u'=-2x), (v = 7 + x^{2}), (v' = 2x).

Step2: Apply the quotient rule

[ \begin{align*} f'(x)&=\frac{(-2x)(7 + x^{2})-(3 - x^{2})(2x)}{(7 + x^{2})^{2}}\ &=\frac{-14x-2x^{3}-6x + 2x^{3}}{(7 + x^{2})^{2}}\ &=\frac{-20x}{(7 + x^{2})^{2}} \end{align*} ]

Answer:

(\frac{-20x}{(7 + x^{2})^{2}})