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${:[f^(')(x)=-(320)/(x^(6))" and "f(1)=30.],[f(-2)=]:}$

$\begin{array}{l}f^{\prime}(x)=-\frac{320}{x^{6}} \text { and } f(1)=30 . \\ f(-2)=\end{array}$

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Find derivatives of sine and cosine functions

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Q.

\begin{array}{l}f^{\prime}(x)=-\frac{320}{x^{6}} \text { and } f(1)=30 . \\ f(-2)=\end{array}

Integrate $f'(x)$ : To find $f(-2)$ , we need to integrate $f'(x)$ to get $f(x)$ . Let's integrate $f'(x) = -\frac{320}{x^{6}}$ . $\int f'(x) \, dx = \int -\frac{320}{x^{6}} \, dx$ $f(x) = \int -320x^{-6} \, dx$ $f(x) = \frac{320}{5} * x^{-5} + C$ , where $C$ is the constant of integration.
Find Constant C: Now we use the given $f(1) = 30$ to find the constant C. $30 = \frac{320}{5} \times 1^{-5} + C$ $30 = \frac{320}{5} + C$ $C = 30 - \frac{320}{5}$ $C = 30 - 64$ $C = -34$
Write Function f(x): With C found, we can write the function f(x) as: $\newline$ $f(x) = \frac{320}{5} \times x^{-5} - 34$
Find $f(-2)$ : Now we can find $f(-2)$ by plugging in $x = -2$ into the function $f(x)$ .
$f(-2) = \frac{320}{5} \times (-2)^{-5} - 34$
$f(-2) = \frac{320}{5} \times \frac{1}{(-2)^5} - 34$
$f(-2) = \frac{320}{5} \times \frac{1}{-32} - 34$
$f(-2) = -\frac{320}{(5\times32)} - 34$
$f(-2) = -2 - 34$
$f(-2) = -36$

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