Question

if ( f(x)=\frac{1}{x^{7}} ), then ( f^{prime}(x)= ) a ( \frac{1}{7 x^{8}} ) b ( -\frac{7}{x^{6}} ) c ( -\frac{1}{7 x^{8}} ) d ( -\frac{7}{x^{8}} )

Explanation:

Step1: Rewrite the function

Rewrite \( f(x)=\frac{1}{x^{7}} \) as \( f(x)=x^{-7} \) using the rule \( \frac{1}{a^{n}}=a^{-n} \).

Step2: Apply the power rule

The power rule for differentiation is \( \frac{d}{dx}(x^{n})=nx^{n - 1} \).
For \( f(x)=x^{-7} \), we have \( f^{\prime}(x)=-7x^{-7-1} \).

Step3: Simplify the expression

Simplify \( -7x^{-8} \) to \( -\frac{7}{x^{8}} \) using the rule \( a^{-n}=\frac{1}{a^{n}} \).

Answer:

D. \( -\frac{7}{x^{8}} \)