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Identical expressions
sqrt(x)*exp^(sqrt(x))
square root of (x) multiply by exponent of to the power of ( square root of (x))
√(x)*exp^(√(x))
sqrt(x)*exp(sqrt(x))
sqrtx*expsqrtx
sqrt(x)exp^(sqrt(x))
sqrt(x)exp(sqrt(x))
sqrtxexpsqrtx
sqrtxexp^sqrtx

The derivative of the function/ sqrt(x)*exp^(sqrt(x))

Derivative of sqrt(x)*exp^(sqrt(x))

The solution

You have entered [src]

         ___
  ___  \/ x 
\/ x *E

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

sqrt(x)*E^(sqrt(x))

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
$g{\left(x \right)} = e^{\sqrt{x}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sqrt{x}$ .
2. The derivative of $e^{u}$ is itself.
3. 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 result is: $\frac{e^{\sqrt{x}}}{2} + \frac{e^{\sqrt{x}}}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   ___       ___
 \/ x      \/ x 
e         e     
------ + -------
  2          ___
         2*\/ x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of sqrt(x)*exp^(sqrt(x))

The graph:

Piecewise:

The solution