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Derivative of:
Derivative of e^x+1
Derivative of sec(x)
Derivative of e^x*sqrt(x)
Derivative of y(x)=sqrt(3x+5)
Integral of d{x}:
e^x*sqrt(x)
Limit of the function:
e^x*sqrt(x)
Identical expressions
e^x*sqrt(x)
e to the power of x multiply by square root of (x)
e^x*√(x)
ex*sqrt(x)
ex*sqrtx
e^xsqrt(x)
exsqrt(x)
exsqrtx
e^xsqrtx

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

Derivative of e^x*sqrt(x)

The solution

You have entered [src]

 x   ___
E *\/ x

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

E^x*sqrt(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/  ___     3         3        3   \  x
|\/ x  - ------ + ------- + ------|*e 
|           3/2       ___      5/2|   
\        4*x      2*\/ x    8*x   /

$$\left(\sqrt{x} + \frac{3}{2 \sqrt{x}} - \frac{3}{4 x^{\frac{3}{2}}} + \frac{3}{8 x^{\frac{5}{2}}}\right) e^{x}$$

Simplify

The graph

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Identical expressions

Derivative of e^x*sqrt(x)

The graph:

Piecewise:

The solution