Answer :

To find the derivative of [tex]\( y = e^{x^{\frac{1}{2}}} \)[/tex] with respect to [tex]\( x \)[/tex], we will use the chain rule. The chain rule is a formula to compute the derivative of a composite function.

The function [tex]\( y = e^{x^{\frac{1}{2}}} \)[/tex] is a composite function where the outer function is the exponential function and the inner function is [tex]\( u = x^{\frac{1}{2}} \)[/tex].

Let's go through the solution step-by-step:

1. Identify the inner function and the outer function:
- Inner function: [tex]\( u = x^{\frac{1}{2}} \)[/tex]
- Outer function: [tex]\( y = e^u \)[/tex]

2. Differentiate the outer function with respect to the inner function [tex]\( u \)[/tex]:
[tex]\[ \frac{d y}{d u} = \frac{d}{d u} (e^u) = e^u \][/tex]

3. Differentiate the inner function with respect to [tex]\( x \)[/tex]:
[tex]\[ \frac{d u}{d x} = \frac{d}{d x} (x^{\frac{1}{2}}) = \frac{1}{2} x^{-\frac{1}{2}} \][/tex]

4. Apply the chain rule:
The chain rule states that:
[tex]\[ \frac{d y}{d x} = \frac{d y}{d u} \cdot \frac{d u}{d x} \][/tex]

5. Substitute the derivatives from steps 2 and 3:
[tex]\[ \frac{d y}{d x} = e^u \cdot \frac{1}{2} x^{-\frac{1}{2}} \][/tex]

6. Substitute back the expression for [tex]\( u \)[/tex] (which is [tex]\( x^{\frac{1}{2}} \)[/tex]):
[tex]\[ \frac{d y}{d x} = e^{x^{\frac{1}{2}}} \cdot \frac{1}{2} x^{-\frac{1}{2}} \][/tex]

7. Simplify the expression:
[tex]\[ \frac{d y}{d x} = \frac{e^{x^{\frac{1}{2}}}}{2 x^{\frac{1}{2}}} \][/tex]

8. Final answer:
[tex]\[ \frac{d y}{d x} = \frac{1}{2} \cdot \frac{e^{x^{\frac{1}{2}}}}{x^{\frac{1}{2}}} \][/tex]

Thus, the derivative of [tex]\( y = e^{x^{\frac{1}{2}}} \)[/tex] with respect to [tex]\( x \)[/tex] is:
[tex]\[ \frac{d y}{d x} = \frac{1}{2} \cdot \frac{e^{x^{\frac{1}{2}}}}{x^{\frac{1}{2}}} \][/tex]