Question

if ( f(x)=4+\frac{7}{x}+\frac{6}{x^{2}} ), find ( f^{prime}(x) ).
find ( f^{prime}(4) ).
find ( f^{prime prime}(x) ).
find ( f^{prime prime}(4) ).

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=4 + \frac{7}{x}+\frac{6}{x^{2}}\) as \(f(x)=4 + 7x^{-1}+6x^{-2}\).

Step2: Find the first - derivative \(f^{\prime}(x)\)

Using the power rule \((x^{n})^\prime=nx^{n - 1}\), we have:
\(f^{\prime}(x)=0+7\times(-1)x^{-2}+6\times(-2)x^{-3}=-\frac{7}{x^{2}}-\frac{12}{x^{3}}\).

Step3: Find \(f^{\prime}(4)\)

Substitute \(x = 4\) into \(f^{\prime}(x)\):
\(f^{\prime}(4)=-\frac{7}{4^{2}}-\frac{12}{4^{3}}=-\frac{7}{16}-\frac{12}{64}=-\frac{28 + 12}{64}=-\frac{40}{64}=-\frac{5}{8}\).

Step4: Find the second - derivative \(f^{\prime\prime}(x)\)

Differentiate \(f^{\prime}(x)=-7x^{-2}-12x^{-3}\) using the power rule.
\(f^{\prime\prime}(x)=(-7)\times(-2)x^{-3}+(-12)\times(-3)x^{-4}=\frac{14}{x^{3}}+\frac{36}{x^{4}}\).

Step5: Find \(f^{\prime\prime}(4)\)

Substitute \(x = 4\) into \(f^{\prime\prime}(x)\):
\(f^{\prime\prime}(4)=\frac{14}{4^{3}}+\frac{36}{4^{4}}=\frac{14}{64}+\frac{36}{256}=\frac{56 + 36}{256}=\frac{92}{256}=\frac{23}{64}\).

Answer:

\(f^{\prime}(4)=-\frac{5}{8}\), \(f^{\prime\prime}(x)=\frac{14}{x^{3}}+\frac{36}{x^{4}}\), \(f^{\prime\prime}(4)=\frac{23}{64}\)