Question

if ( f(x)=\frac{3 x^{2}+7 x+3}{sqrt{x}} ), then:
( f^{prime}(x)= )
( f^{prime}(1)= )
question help: video message instructor

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:

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