Question

1. if ( f(x)=3 e^{x}+pi^{3}-e^{4} ),

(a) ( 3 e^{x}+3 pi^{2}-4 e^{3} )

1. if ( f(x)=\frac{1}{3}left(x^{3}-\frac{1}{x^{3}}

ight) ), then ( f^{prime}(-1)= )
(a) -2 (b) 0 (c) 2 (d) 4

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