if ( f(x)=\frac{1}{x^{7}} ), then ( f^{prime}(x)= ) \na ( \frac{1}{7 x^{3}} ) \nb ( -\frac{7}{x^{6}} ) \nc (…

if ( f(x)=\frac{1}{x^{7}} ), then ( f^{prime}(x)= ) \na ( \frac{1}{7 x^{3}} ) \nb ( -\frac{7}{x^{6}} ) \nc ( -\frac{1}{7 x^{8}} ) \nd ( -\frac{7}{x^{8}} )

if ( f(x)=\frac{1}{x^{7}} ), then ( f^{prime}(x)= ) \na ( \frac{1}{7 x^{3}} ) \nb ( -\frac{7}{x^{6}} ) \nc ( -\frac{1}{7 x^{8}} ) \nd ( -\frac{7}{x^{8}} )

Answer

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=\frac{1}{x^{7}} ) as ( f(x)=x^{-7} ).

Step2: Apply the power rule

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

Step3: Simplify the expression

( f^{\prime}(x)=-7x^{-8}=-\frac{7}{x^{8}} ).

Answer:

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