suppose that ( f(x) = sqrt{3 + e^x} ). find ( f(2) ).

Explanation:

Step1: Rewrite the function

Rewrite ( f(x)=\sqrt{3 + e^{x}} ) as ( f(x)=(3 + e^{x})^{\frac{1}{2}} ).

Step2: Apply the chain rule

The chain rule states that if ( y = u^n ) and ( u = g(x) ), then ( y'=n\cdot u^{n - 1}\cdot u' ). Let ( u = 3+e^{x} ), so ( n=\frac{1}{2} ), ( u'=e^{x} ). Then ( f'(x)=\frac{1}{2}(3 + e^{x})^{-\frac{1}{2}}\cdot e^{x}=\frac{e^{x}}{2\sqrt{3 + e^{x}}} ).

Step3: Substitute ( x = 2 )

Substitute ( x = 2 ) into ( f'(x) ), we get ( f'(2)=\frac{e^{2}}{2\sqrt{3 + e^{2}}} ).

Answer:

(\frac{e^{2}}{2\sqrt{3 + e^{2}}})