Question

differentiate $f(x)=\frac{2}{x^{5}} - 5$

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\frac{2}{x^{5}} - 5\) as \(f(x)=2x^{- 5}-5\).

Step2: Apply the power rule

The power rule is \(\frac{d}{dx}(x^{n})=nx^{n - 1}\).
For the term \(2x^{-5}\), using the power rule: \(\frac{d}{dx}(2x^{-5})=2\times(-5)x^{-5 - 1}=-10x^{-6}\).
For the constant term \(-5\), since \(\frac{d}{dx}(C) = 0\) (where \(C\) is a constant), \(\frac{d}{dx}(-5)=0\).

Step3: Combine the derivatives

\(f^{\prime}(x)=\frac{d}{dx}(2x^{-5}-5)=\frac{d}{dx}(2x^{-5})-\frac{d}{dx}(5)=-10x^{-6}=-\frac{10}{x^{6}}\)

Answer:

\(-\frac{10}{x^{6}}\)