find f(-4) if f(x)=\frac{x^{4}}{5}-8x.\nf(-4)=square (simplify your answer. type an integer or a fraction.)
find f(-4) if f(x)=\frac{x^{4}}{5}-8x.\nf(-4)=square (simplify your answer. type an integer or a fraction.)
Answer
Explanation:
Step1: Find the derivative of $f(x)$
Use the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $y=\frac{x^{4}}{5}-8x$, the derivative $f'(x)=\frac{4x^{3}}{5}-8$.
Step2: Substitute $x = - 4$ into $f'(x)$
$f'(-4)=\frac{4(-4)^{3}}{5}-8$. First, calculate $(-4)^{3}=-64$. Then $\frac{4(-4)^{3}}{5}=\frac{4\times(-64)}{5}=-\frac{256}{5}$. So $f'(-4)=-\frac{256}{5}-8$. Rewrite $8$ as $\frac{40}{5}$, then $f'(-4)=-\frac{256}{5}-\frac{40}{5}=-\frac{256 + 40}{5}=-\frac{296}{5}$.
Answer:
$-\frac{296}{5}$