if ( f(x)=5+\frac{6}{x}+\frac{3}{x^{2}} ), find ( f^{prime}(x) ).\nfind ( f^{prime}(5) ).\nfind ( f^{prime…

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=5 + \frac{6}{x}+\frac{3}{x^{2}}) as (f(x)=5 + 6x^{-1}+3x^{-2}).

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

Use the power rule ((x^{n})^\prime=nx^{n - 1}). For (y = 5), since the derivative of a constant (C) is (0) ((C^\prime=0)), for (y = 6x^{-1}), (y^\prime=6\times(-1)x^{-1 - 1}=-6x^{-2}), for (y = 3x^{-2}), (y^\prime=3\times(-2)x^{-2 - 1}=-6x^{-3}). So (f^{\prime}(x)=-6x^{-2}-6x^{-3}=-\frac{6}{x^{2}}-\frac{6}{x^{3}}).

Step3: Find (f^{\prime}(5))

Substitute (x = 5) into (f^{\prime}(x)). (f^{\prime}(5)=-\frac{6}{5^{2}}-\frac{6}{5^{3}}=-\frac{6\times5}{5^{3}}-\frac{6}{5^{3}}=-\frac{30 + 6}{125}=-\frac{36}{125}).

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

Differentiate (f^{\prime}(x)=-6x^{-2}-6x^{-3}) using the power rule. For (y=-6x^{-2}), (y^\prime=-6\times(-2)x^{-2 - 1}=12x^{-3}), for (y=-6x^{-3}), (y^\prime=-6\times(-3)x^{-3 - 1}=18x^{-4}). So (f^{\prime\prime}(x)=12x^{-3}+18x^{-4}=\frac{12}{x^{3}}+\frac{18}{x^{4}}).

Step5: Find (f^{\prime\prime}(5))

Substitute (x = 5) into (f^{\prime\prime}(x)). (f^{\prime\prime}(5)=\frac{12}{5^{3}}+\frac{18}{5^{4}}=\frac{12\times5}{5^{4}}+\frac{18}{5^{4}}=\frac{60 + 18}{625}=\frac{78}{625}).

Answer:

(f^{\prime}(x)=-\frac{6}{x^{2}}-\frac{6}{x^{3}}) (f^{\prime}(5)=-\frac{36}{125}) (f^{\prime\prime}(x)=\frac{12}{x^{3}}+\frac{18}{x^{4}}) (f^{\prime\prime}(5)=\frac{78}{625})