Derivative of exp^sqrtx

The solution

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 \/ x 
E

$$e^{\sqrt{x}}$$

E^(sqrt(x))

Detail solution

Let $u = \sqrt{x}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{x}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
The result of the chain rule is:

$\frac{e^{\sqrt{x}}}{2 \sqrt{x}}$

The answer is:

$\frac{e^{\sqrt{x}}}{2 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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  \/ x 
 e     
-------
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2*\/ x

$$\frac{e^{\sqrt{x}}}{2 \sqrt{x}}$$

Simplify

The second derivative [src]

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/1    1  \  \/ x 
|- - ----|*e     
|x    3/2|       
\    x   /       
-----------------
        4

$$\frac{\left(\frac{1}{x} - \frac{1}{x^{\frac{3}{2}}}\right) e^{\sqrt{x}}}{4}$$

Simplify

The third derivative [src]

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/ 1     3     3  \  \/ x 
|---- - -- + ----|*e     
| 3/2    2    5/2|       
\x      x    x   /       
-------------------------
            8

$$\frac{\left(- \frac{3}{x^{2}} + \frac{1}{x^{\frac{3}{2}}} + \frac{3}{x^{\frac{5}{2}}}\right) e^{\sqrt{x}}}{8}$$

Simplify

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Identical expressions

Derivative of exp^sqrtx

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The solution