if ( f(x)=\frac{3 x^{2}+7 x+3}{sqrt{x}} ), then:\n( f^{prime}(x)= )\n( f^{prime}(1)= )\nquestion help: video…

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=\frac{3x^{2}+7x + 3}{\sqrt{x}}=3x^{\frac{3}{2}}+7x^{\frac{1}{2}}+3x^{-\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 ((x^{n})^\prime=nx^{n - 1}). For (y = 3x^{\frac{3}{2}}), (y^\prime=3\times\frac{3}{2}x^{\frac{3}{2}-1}=\frac{9}{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 = 3x^{-\frac{1}{2}}), (y^\prime=3\times(-\frac{1}{2})x^{-\frac{1}{2}-1}=-\frac{3}{2}x^{-\frac{3}{2}}). So (f^\prime(x)=\frac{9}{2}x^{\frac{1}{2}}+\frac{7}{2}x^{-\frac{1}{2}}-\frac{3}{2}x^{-\frac{3}{2}}).

Step3: Simplify (f^\prime(x))

(f^\prime(x)=\frac{9x^{\frac{3}{2}}+7x - 3}{2x^{\frac{3}{2}}}).

Step4: Evaluate (f^\prime(1))

Substitute (x = 1) into (f^\prime(x)). (f^\prime(1)=\frac{9\times1^{\frac{3}{2}}+7\times1 - 3}{2\times1^{\frac{3}{2}}}=\frac{9 + 7-3}{2}=\frac{13}{2}).

Answer:

(f^\prime(x)=\frac{9x^{\frac{3}{2}}+7x - 3}{2x^{\frac{3}{2}}}), (f^\prime(1)=\frac{13}{2})