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