Question

the derivative of ( f(x)=\frac{8}{2 x^{5}} ) is ( f^{prime}(x)=\frac{b}{x^{k}} ).
find the value of ( b ). give an exact answer as an integer.

Explanation:

Step1: Simplify the function

First, simplify \( f(x)=\frac{8}{2x^{5}} = \frac{4}{x^{5}}=4x^{- 5}\).

Step2: Apply the power rule

The power rule for differentiation is \((x^{n})^\prime=nx^{n - 1}\).
For \(y = ax^{n}\), \(y^\prime=a\times n\times x^{n - 1}\).
Here \(a = 4\) and \(n=-5\), so \(f^\prime(x)=4\times(-5)x^{-5 - 1}\).
\(f^\prime(x)=-20x^{-6}=\frac{-20}{x^{6}}\).

Answer:

\(-20\)