Question

if $f(x)=\frac{7 - x^{2}}{5 + x^{2}}$, find: $f(x)=$

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, $u = 7 - x^{2}$, $v=5 + x^{2}$.

Step2: Find $u'$ and $v'$

Differentiate $u = 7 - x^{2}$ with respect to $x$. Using the power - rule $\frac{d}{dx}(x^{n})=nx^{n - 1}$ and $\frac{d}{dx}(c)=0$ (where $c$ is a constant), we get $u'=-2x$. Differentiate $v = 5 + x^{2}$ with respect to $x$, we get $v' = 2x$.

Step3: Apply the quotient - rule

Substitute $u$, $v$, $u'$, and $v'$ into the quotient - rule formula:
\[

$$\begin{align*} f'(x)&=\frac{(-2x)(5 + x^{2})-(7 - x^{2})(2x)}{(5 + x^{2})^{2}}\\ &=\frac{-10x-2x^{3}-14x + 2x^{3}}{(5 + x^{2})^{2}}\\ &=\frac{-24x}{(5 + x^{2})^{2}} \end{align*}$$

\]

Answer:

$\frac{-24x}{(5 + x^{2})^{2}}$