Question

homework 3.3 differentiation rules
score: 90/120 answered: 9/12
question 10
textbook videos -
if (f(x)=\frac{sqrt{x}-5}{sqrt{x}+5}), find:
(f(x)=)
(f(2)=)
question help: video

Explanation:

Step1: Use quotient - rule

The quotient - rule states that if $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = \sqrt{x}-5=x^{\frac{1}{2}}-5$, $u^\prime=\frac{1}{2}x^{-\frac{1}{2}}$, $v=\sqrt{x}+5=x^{\frac{1}{2}}+5$, and $v^\prime=\frac{1}{2}x^{-\frac{1}{2}}$.

Step2: Substitute into quotient - rule formula

\[

$$\begin{align*} f^\prime(x)&=\frac{(\frac{1}{2}x^{-\frac{1}{2}})(x^{\frac{1}{2}} + 5)-(x^{\frac{1}{2}}-5)(\frac{1}{2}x^{-\frac{1}{2}})}{(x^{\frac{1}{2}}+5)^{2}}\\ &=\frac{\frac{1}{2}x^{-\frac{1}{2}+\frac{1}{2}}+\frac{5}{2}x^{-\frac{1}{2}}-\frac{1}{2}x^{-\frac{1}{2}+\frac{1}{2}}+\frac{5}{2}x^{-\frac{1}{2}}}{(x^{\frac{1}{2}} + 5)^{2}}\\ &=\frac{\frac{1}{2}+\frac{5}{2}x^{-\frac{1}{2}}-\frac{1}{2}+\frac{5}{2}x^{-\frac{1}{2}}}{(x^{\frac{1}{2}}+5)^{2}}\\ &=\frac{5x^{-\frac{1}{2}}}{(x^{\frac{1}{2}} + 5)^{2}}=\frac{5}{ \sqrt{x}( \sqrt{x}+5)^{2}} \end{align*}$$

\]

Step3: Find $f^\prime(2)$

Substitute $x = 2$ into $f^\prime(x)$:
\[

$$\begin{align*} f^\prime(2)&=\frac{5}{\sqrt{2}(\sqrt{2}+5)^{2}}\\ &=\frac{5}{\sqrt{2}(2 + 10\sqrt{2}+25)}\\ &=\frac{5}{\sqrt{2}(27 + 10\sqrt{2})}\\ &=\frac{5}{27\sqrt{2}+20} \end{align*}$$

\]

Answer:

$f^\prime(x)=\frac{5}{\sqrt{x}(\sqrt{x}+5)^{2}}$, $f^\prime(2)=\frac{5}{27\sqrt{2}+20}$