70 - 71. higher - order derivatives find f(x), f(x), and f(x).\n70. f(x)=\frac{1}{x}\n71. f(x)=x^{2}(2 +…

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=\frac{1}{x}=x^{- 1}$.

Step2: Find the first - derivative

Using the power rule $(x^n)^\prime=nx^{n - 1}$, we have $f^\prime(x)=-1\times x^{-1 - 1}=-x^{-2}=-\frac{1}{x^{2}}$.

Step3: Find the second - derivative

Differentiate $f^\prime(x)=-x^{-2}$ again using the power rule. $f^{\prime\prime}(x)=-(-2)x^{-2 - 1}=2x^{-3}=\frac{2}{x^{3}}$.

Step4: Find the third - derivative

Differentiate $f^{\prime\prime}(x)=2x^{-3}$ using the power rule. $f^{\prime\prime\prime}(x)=2\times(-3)x^{-3 - 1}=-6x^{-4}=-\frac{6}{x^{4}}$.

Answer:

$f^\prime(x)=-\frac{1}{x^{2}}$, $f^{\prime\prime}(x)=\frac{2}{x^{3}}$, $f^{\prime\prime\prime}(x)=-\frac{6}{x^{4}}$