7. if ( f(x)=\frac{1}{3}left(x^{3}-\frac{1}{x^{3}}\right) ), then ( f^{prime}(-1)= ) (a) ( -2 ) (b) ( 0 )…

Explanation:

Step1: Simplify the function

Rewrite (\frac{1}{x^{3}}) as (x^{-3}). So (f(x)=\frac{1}{3}(x^{3}-x^{-3})).

Step2: Differentiate using the power rule

The power rule is ((x^{n})^\prime = nx^{n - 1}). Differentiate (f(x)): (f^\prime(x)=\frac{1}{3}(3x^{2}+ 3x^{-4})) (since ((x^{3})^\prime=3x^{2}) and ((-x^{-3})^\prime=(-1)\times(-3)x^{-3 - 1}=3x^{-4})). Simplify (f^\prime(x)) to (f^\prime(x)=x^{2}+x^{-4}).

Step3: Substitute (x = - 1)

When (x=-1), (f^\prime(-1)=(-1)^{2}+(-1)^{-4}). Since ((-1)^{2}=1) and ((-1)^{-4}=\frac{1}{(-1)^{4}} = 1). Then (f^\prime(-1)=1 + 1).

Answer:

C. (2)