exercsies 3.3 differentiation rules\nscore: 31/40 answered: 16/19\nquestion 17\ntextbook videos +\nif (…

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} )