Question

score on last try: 5 of 10 pts. see details for more. at least one scored part is incorrect. jump to first changable incorrect part. > next question get a similar question you can retry this question below if $f(x) = 3 + \frac{8}{x} + \frac{2}{x^2}$, find $f(x)$. find $f(4)$. $-\frac{9}{16}$

Explanation:

Step1: Rewrite the function

Rewrite \( f(x) = 3+\frac{8}{x}+\frac{2}{x^{2}} \) as \( f(x)=3 + 8x^{-1}+2x^{-2} \).

Step2: Differentiate term by term

Using the power rule \( \frac{d}{dx}(x^{n})=nx^{n - 1} \) and the constant rule \( \frac{d}{dx}(c)=0 \) (where \( c \) is a constant):

• The derivative of \( 3 \) (a constant) is \( 0 \).
• The derivative of \( 8x^{-1} \) is \( 8\times(- 1)x^{-1-1}=-8x^{-2} \).
• The derivative of \( 2x^{-2} \) is \( 2\times(-2)x^{-2 - 1}=-4x^{-3} \).

So, \( f^{\prime}(x)=0-8x^{-2}-4x^{-3} \), which can be rewritten as \( f^{\prime}(x)=-\frac{8}{x^{2}}-\frac{4}{x^{3}} \).

Answer:

\( f^{\prime}(x)=-\frac{8}{x^{2}}-\frac{4}{x^{3}} \) (or equivalent forms like \( - 8x^{-2}-4x^{-3} \))