Question

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:

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