consider f(x) = x^2 / (5 + x)^5. a) find f(x) using the quotient rule and the extended power rule. b) note…

Explanation:

Step1: Recall Quotient Rule

The Quotient Rule states that if $y = \frac{u}{v}$, then $y'=\frac{u'v - uv'}{v^{2}}$. Here, $u = x^{2}$, so $u'=2x$, and $v=(5 + x)^{5}$, so $v' = 5(5 + x)^{4}$ by the Extended - Power Rule ($(u^{n})'=nu^{n - 1}u'$ with $u = 5 + x$ and $n = 5$).

Step2: Apply Quotient Rule

$f'(x)=\frac{(2x)(5 + x)^{5}-x^{2}\times5(5 + x)^{4}}{(5 + x)^{10}}$.

Step3: Simplify the expression

Factor out $(5 + x)^{4}$ from the numerator: [ \begin{align*} f'(x)&=\frac{(5 + x)^{4}[2x(5 + x)-5x^{2}]}{(5 + x)^{10}}\ &=\frac{10x+2x^{2}-5x^{2}}{(5 + x)^{6}}\ &=\frac{10x - 3x^{2}}{(5 + x)^{6}} \end{align*} ]

Step4: Recall Product Rule for part (b)

The Product Rule states that if $y = uv$, then $y'=u'v+uv'$. Here, $u = x^{2}$, $u' = 2x$, $v=(5 + x)^{-5}$, and $v'=- 5(5 + x)^{-6}$ by the Extended - Power Rule.

Step5: Apply Product Rule

$f'(x)=(2x)(5 + x)^{-5}+x^{2}\times(-5)(5 + x)^{-6}$.

Step6: Simplify the expression for part (b)

Get a common denominator of $(5 + x)^{6}$: [ \begin{align*} f'(x)&=\frac{2x(5 + x)-5x^{2}}{(5 + x)^{6}}\ &=\frac{10x+2x^{2}-5x^{2}}{(5 + x)^{6}}\ &=\frac{10x - 3x^{2}}{(5 + x)^{6}} \end{align*} ]

Step7: Compare answers

The answers from part (a) and part (b) are the same.

Answer:

a) $f'(x)=\frac{10x - 3x^{2}}{(5 + x)^{6}}$ b) $f'(x)=\frac{10x - 3x^{2}}{(5 + x)^{6}}$ c) The answers are the same.