6. if ( f(x)=3 e^{x}+pi^{3}-e^{4} ),\n(a) ( 3 e^{x}+3 pi^{2}-4 e^{3} )\n7. if ( f(x)=\frac{1}{3}left(x^{3}-\f…

6. if ( f(x)=3 e^{x}+pi^{3}-e^{4} ),\n(a) ( 3 e^{x}+3 pi^{2}-4 e^{3} )\n7. if ( f(x)=\frac{1}{3}left(x^{3}-\frac{1}{x^{3}}\right) ), then ( f^{prime}(-1)= )\n(a) -2 (b) 0 (c) 2 (d) 4

6. if ( f(x)=3 e^{x}+pi^{3}-e^{4} ),\n(a) ( 3 e^{x}+3 pi^{2}-4 e^{3} )\n7. if ( f(x)=\frac{1}{3}left(x^{3}-\frac{1}{x^{3}}\right) ), then ( f^{prime}(-1)= )\n(a) -2 (b) 0 (c) 2 (d) 4

Answer

Explanation:

Step1: Differentiate the function

Use the power rule ((x^n)^\prime = nx^{n - 1}) and the rule ((e^x)^\prime=e^x). For (y = f(x)=\frac{1}{3}(x^{3}-\frac{1}{x^{3}})=\frac{1}{3}(x^{3}-x^{- 3})), then (f^\prime(x)=\frac{1}{3}(3x^{2}+3x^{-4})) (by applying ((x^n)^\prime=nx^{n - 1}), ((ax^n)^\prime = anx^{n - 1})).

Step2: Substitute (x=-1) into the derivative

Substitute (x = - 1) into (f^\prime(x)). (f^\prime(-1)=\frac{1}{3}(3\times(-1)^{2}+3\times(-1)^{-4})) First, calculate ((-1)^{2}=1) and ((-1)^{-4}=\frac{1}{(-1)^4}=1). Then (f^\prime(-1)=\frac{1}{3}(3\times1 + 3\times1)=\frac{1}{3}(3 + 3)=2).

Answer:

C. 2