Question

exercsies 3.3 differentiation rules
score: 31/40 answered: 16/19
question 17
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if ( f(x)=\frac{2 x^{2}+3 x + 5}{sqrt{x}} ), then:
( f^{prime}(x)= )
( f^{prime}(2)= )
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Explanation:

Step1: Simplify the function

Rewrite \( f(x)=\frac{2x^{2}+3x + 5}{\sqrt{x}} \) as \( f(x)=2x^{\frac{3}{2}}+3x^{\frac{1}{2}}+5x^{-\frac{1}{2}} \) using the rule \( \frac{a^{m}}{a^{n}}=a^{m - n} \) (\(a=x\), \(m = 2,1,0\) and \(n=\frac{1}{2}\)).

Step2: Differentiate term - by - term

Use the power rule \( \frac{d}{dx}(x^{n})=nx^{n - 1} \).
For \( y = 2x^{\frac{3}{2}} \), \( y^\prime=2\times\frac{3}{2}x^{\frac{3}{2}-1}=3x^{\frac{1}{2}} \).
For \( y = 3x^{\frac{1}{2}} \), \( y^\prime=3\times\frac{1}{2}x^{\frac{1}{2}-1}=\frac{3}{2}x^{-\frac{1}{2}} \).
For \( y = 5x^{-\frac{1}{2}} \), \( y^\prime=5\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-\frac{5}{2}x^{-\frac{3}{2}} \).
So, \( f^\prime(x)=3x^{\frac{1}{2}}+\frac{3}{2}x^{-\frac{1}{2}}-\frac{5}{2}x^{-\frac{3}{2}} \).

Step3: Evaluate \( f^\prime(2) \)

Substitute \( x = 2 \) into \( f^\prime(x) \).
\( f^\prime(2)=3\sqrt{2}+\frac{3}{2\sqrt{2}}-\frac{5}{2\times(\sqrt{2})^{3}} \).
Rationalize the denominators:
\( f^\prime(2)=3\sqrt{2}+\frac{3\sqrt{2}}{4}-\frac{5\sqrt{2}}{8} \).
Find a common denominator (8):
\( f^\prime(2)=\frac{24\sqrt{2}+6\sqrt{2}-5\sqrt{2}}{8}=\frac{25\sqrt{2}}{8} \).

Answer:

\( f^\prime(x)=3x^{\frac{1}{2}}+\frac{3}{2}x^{-\frac{1}{2}}-\frac{5}{2}x^{-\frac{3}{2}} \); \( f^\prime(2)=\frac{25\sqrt{2}}{8} \)