Question

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

Explanation:

Step1: Rewrite the function

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

Step2: Apply the chain rule

The chain rule states that if \( y = u^n \) and \( u = g(x) \), then \( y^\prime=n\cdot u^{n - 1}\cdot g^\prime(x) \). Here, \( u = 3+e^{x} \), \( n=\frac{1}{2} \), and the derivative of \( u \) with respect to \( x \) is \( u^\prime=e^{x} \) (since the derivative of \( 3 \) is \( 0 \) and the derivative of \( e^{x} \) is \( e^{x} \)).

So, \( f^\prime(x)=\frac{1}{2}(3 + e^{x})^{\frac{1}{2}-1}\cdot e^{x}=\frac{1}{2}(3 + e^{x})^{-\frac{1}{2}}\cdot e^{x}=\frac{e^{x}}{2\sqrt{3 + e^{x}}} \).

Step3: Evaluate at \( x = 2 \)

Substitute \( x = 2 \) into the derivative: \( f^\prime(2)=\frac{e^{2}}{2\sqrt{3 + e^{2}}} \).

Answer:

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