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

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})