Question

chain rule: problem 1 (1 point) results for this submission 1 of the answers is not correct. let $f(x)=sqrt{\frac{5x + 2}{5x^{2}+4x + 2}}$, $f(x)=sqrt{\frac{5x + 2}{5x^{2}+4x + 2}}$, $f(2)=\frac{4sqrt{22}}{11}$. note: you can earn partial credit on this problem. preview my answers submit answers your score was recorded. your score was successfully sent to canvas. you have attempted this problem 2 times. you received a score of 50% for this attempt. your overall recorded score is 50%. you have unlimited attempts remaining. email instructor

Explanation:

Step1: Recall chain - rule formula

The chain - rule states that if \(y = f(u)\) and \(u = g(x)\), then \(\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}\). For \(y = f(x)=\sqrt{5x^{2}+4x + 2}=(5x^{2}+4x + 2)^{\frac{1}{2}}\), let \(u = 5x^{2}+4x + 2\), so \(y = u^{\frac{1}{2}}\).

Step2: Differentiate \(y\) with respect to \(u\) and \(u\) with respect to \(x\)

First, \(\frac{dy}{du}=\frac{1}{2}u^{-\frac{1}{2}}\) (using the power - rule \(\frac{d}{du}(u^{n})=nu^{n - 1}\) with \(n=\frac{1}{2}\)). Second, \(\frac{du}{dx}=10x + 4\) (using the power - rule \(\frac{d}{dx}(ax^{n})=nax^{n - 1}\) for each term of \(u\)).

Step3: Apply the chain - rule

\(f^{\prime}(x)=\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}=\frac{1}{2}(5x^{2}+4x + 2)^{-\frac{1}{2}}\cdot(10x + 4)=\frac{10x + 4}{2\sqrt{5x^{2}+4x + 2}}=\frac{5x+2}{\sqrt{5x^{2}+4x + 2}}\).

Step4: Evaluate \(f^{\prime}(x)\) at \(x = 2\)

Substitute \(x = 2\) into \(f^{\prime}(x)\). When \(x = 2\), \(5x^{2}+4x + 2=5\times2^{2}+4\times2 + 2=20 + 8+2=30\), and \(5x + 2=5\times2+2 = 12\). So \(f^{\prime}(2)=\frac{5\times2 + 2}{\sqrt{5\times2^{2}+4\times2 + 2}}=\frac{12}{\sqrt{30}}=\frac{12}{\sqrt{6\times5}}=\frac{12}{\sqrt{6}\times\sqrt{5}}=\frac{12\sqrt{30}}{30}=\frac{4\sqrt{30}}{10}=\frac{2\sqrt{30}}{5}
eq\frac{4\sqrt{22}}{11}\). The correct derivative evaluation:
\[

$$\begin{align*} f^{\prime}(x)&=\frac{5x + 2}{\sqrt{5x^{2}+4x + 2}}\\ f^{\prime}(2)&=\frac{5\times2+2}{\sqrt{5\times2^{2}+4\times2 + 2}}=\frac{10 + 2}{\sqrt{20+8 + 2}}=\frac{12}{\sqrt{30}}=\frac{2\sqrt{30}}{5} \end{align*}$$

\]

Answer:

The correct derivative \(f^{\prime}(x)=\frac{5x + 2}{\sqrt{5x^{2}+4x + 2}}\) and \(f^{\prime}(2)=\frac{12}{\sqrt{30}}=\frac{2\sqrt{30}}{5}\)