the derivative of ( f(x)=\frac{8}{2 x^{5}} ) is ( f^{prime}(x)=\frac{b}{x^{k}} ).\nfind the value of ( b )…

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

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

Answer

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)