Question

determine $f(2)$ if $f(x)=24x - \frac{1000}{x^{9}}$.

Explanation:

Step1: Rewrite the function

Rewrite $f(x)=24x - \frac{1000}{x^{9}}$ as $f(x)=24x-1000x^{- 9}$.

Step2: Find the first - derivative

Using the power rule $(x^n)^\prime=nx^{n - 1}$, we have $f^\prime(x)=(24x)^\prime-(1000x^{-9})^\prime=24+9000x^{-10}$.

Step3: Find the second - derivative

Differentiate $f^\prime(x)$ again. $f^{\prime\prime}(x)=(24)^\prime+(9000x^{-10})^\prime=-90000x^{-11}$.

Step4: Find the third - derivative

Differentiate $f^{\prime\prime}(x)$ again. $f^{\prime\prime\prime}(x)=990000x^{-12}=\frac{990000}{x^{12}}$.

Step5: Evaluate $f^{\prime\prime\prime}(2)$

Substitute $x = 2$ into $f^{\prime\prime\prime}(x)$. $f^{\prime\prime\prime}(2)=\frac{990000}{2^{12}}=\frac{990000}{4096}=\frac{123750}{512}=\frac{61875}{256}$.

Answer:

$\frac{61875}{256}$