Question

score on last try: 0 of 1 pts. see detai
at least one scored part is incorrect. ju

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if ( f(x)=\frac{7 x^{2}+7 x + 2}{sqrt{x}} ), then:
( f^{prime}(x)=\frac{left(21 x - x^{-1}+7
ight)}{2 sqrt{x}} )
( f^{prime}(5)=\frac{559 sqrt{5}}{50} )

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\frac{7x^{2}+7x + 2}{\sqrt{x}}=7x^{\frac{3}{2}}+7x^{\frac{1}{2}}+2x^{-\frac{1}{2}}\) using the rule \(\frac{a^{m}}{a^{n}}=a^{m - n}\).

Step2: Differentiate term - by - term

Use the power rule \((x^{n})^\prime=nx^{n - 1}\).
For \(y = 7x^{\frac{3}{2}}\), \(y^\prime=7\times\frac{3}{2}x^{\frac{3}{2}-1}=\frac{21}{2}x^{\frac{1}{2}}\).
For \(y = 7x^{\frac{1}{2}}\), \(y^\prime=7\times\frac{1}{2}x^{\frac{1}{2}-1}=\frac{7}{2}x^{-\frac{1}{2}}\).
For \(y = 2x^{-\frac{1}{2}}\), \(y^\prime=2\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-x^{-\frac{3}{2}}\).
So \(f^\prime(x)=\frac{21}{2}x^{\frac{1}{2}}+\frac{7}{2}x^{-\frac{1}{2}}-x^{-\frac{3}{2}}=\frac{21x - x^{-1}+7}{2\sqrt{x}}\).

Step3: Evaluate \(f^\prime(5)\)

Substitute \(x = 5\) into \(f^\prime(x)\).
\(f^\prime(5)=\frac{21\times5-5^{-1}+7}{2\sqrt{5}}=\frac{105-\frac{1}{5}+7}{2\sqrt{5}}=\frac{\frac{525 - 1+35}{5}}{2\sqrt{5}}=\frac{\frac{559}{5}}{2\sqrt{5}}=\frac{559}{10\sqrt{5}}=\frac{559\sqrt{5}}{50}\).

Answer:

\(f^\prime(x)=\frac{21x - x^{-1}+7}{2\sqrt{x}}\), \(f^\prime(5)=\frac{559\sqrt{5}}{50}\)