if ( f(x)=\frac{7 x^{2}+7 x + 2}{sqrt{x}} ), then:\n( f^{prime}(x)= )\n( f^{prime}(5)= )\nquestion help:

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: Apply the power rule

The power rule is ((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: Calculate (f^\prime(5))

Substitute (x = 5) into (f^\prime(x)). (f^\prime(5)=\frac{21\times5-\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}\approx\frac{559\times2.236}{50}\approx25.0).

Answer:

(f^\prime(x)=\frac{21x - x^{-1}+7}{2\sqrt{x}}); (f^\prime(5)=\frac{559\sqrt{5}}{50}\approx25.0)