exercsies 3.3 differentiation
score: 10/19 answered: 10/19
question 11
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if ( f(x)=\frac{6 - x^{2}}{3 + x^{2}} ), find:
( f^{prime}(x)= )
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Accepted Answer

# Explanation:
## Step1: Apply the quotient rule
The quotient rule states that if $f(x)=\frac{u(x)}{v(x)}$, then $f^{\prime}(x)=\frac{u^{\prime}(x)v(x)-u(x)v^{\prime}(x)}{v(x)^{2}}$. Here, $u = 6 - x^{2}$, so $u^{\prime}=-2x$; $v = 3 + x^{2}$, so $v^{\prime}=2x$.
## Step2: Substitute into the quotient rule formula
$$
\begin{align*}
f^{\prime}(x)&=\frac{(-2x)(3 + x^{2})-(6 - x^{2})(2x)}{(3 + x^{2})^{2}}\
&=\frac{-6x-2x^{3}-12x + 2x^{3}}{(3 + x^{2})^{2}}\
&=\frac{-18x}{(3 + x^{2})^{2}}
\end{align*}
$$

# Answer:
$\frac{-18x}{(3 + x^{2})^{2}}$

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Explanation:

Step1: Apply the quotient rule

Step2: Substitute into the quotient rule formula

Answer: