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Identical expressions
e^sqrt(two *x)*(sqrt(two *x)- one)
e to the power of square root of (2 multiply by x) multiply by ( square root of (2 multiply by x) minus 1)
e to the power of square root of (two multiply by x) multiply by ( square root of (two multiply by x) minus one)
e^√(2*x)*(√(2*x)-1)
esqrt(2*x)*(sqrt(2*x)-1)
esqrt2*x*sqrt2*x-1
e^sqrt(2x)(sqrt(2x)-1)
esqrt(2x)(sqrt(2x)-1)
esqrt2xsqrt2x-1
e^sqrt2xsqrt2x-1
Similar expressions
e^sqrt(2*x)*(sqrt(2*x)+1)

The derivative of the function/ e^sqrt(2*x)*(sqrt(2*x)-1)

Derivative of e^sqrt(2x)(sqrt(2*x)-1)

The solution

You have entered [src]

   _____              
 \/ 2*x  /  _____    \
E       *\\/ 2*x  - 1/

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

E^(sqrt(2*x))*(sqrt(2*x) - 1)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = e^{\sqrt{2 x}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \sqrt{2 x}$ .
2. The derivative of $e^{u}$ is itself.
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sqrt{2 x}$ :
  1. Let $u = 2 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $\frac{\sqrt{2}}{2 \sqrt{x}}$
  The result of the chain rule is:
  
  $\frac{\sqrt{2} e^{\sqrt{2 x}}}{2 \sqrt{x}}$
$g{\left(x \right)} = \sqrt{2 x} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\sqrt{2 x} - 1$ term by term:
  1. Let $u = 2 x$ .
  2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. Apply the power rule: $x$ goes to $1$
      So, the result is: $2$
    The result of the chain rule is:
    
    $\frac{\sqrt{2}}{2 \sqrt{x}}$
  4. The derivative of the constant $-1$ is zero.
  The result is: $\frac{\sqrt{2}}{2 \sqrt{x}}$
The result is: $\frac{\sqrt{2} \left(\sqrt{2 x} - 1\right) e^{\sqrt{2 x}}}{2 \sqrt{x}} + \frac{\sqrt{2} e^{\sqrt{2 x}}}{2 \sqrt{x}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         _____                          _____
  ___  \/ 2*x      ___ /  _____    \  \/ 2*x 
\/ 2 *e          \/ 2 *\\/ 2*x  - 1/*e       
-------------- + ----------------------------
       ___                     ___           
   2*\/ x                  2*\/ x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/                                                                         /      ___\\         
|                                                                     ___ |2   \/ 2 ||         
|                                                                 3*\/ 2 *|- - -----||         
|                          /           ___       ___\       ___           |x     3/2||    _____
|  6    /       ___   ___\ |  6    2*\/ 2    3*\/ 2 |   3*\/ 2            \     x   /|  \/ 2*x 
|- -- + \-1 + \/ 2 *\/ x /*|- -- + ------- + -------| + ------- + -------------------|*e       
|   2                      |   2      3/2       5/2 |      5/2             ___       |         
\  x                       \  x      x         x    /     x              \/ x        /         
-----------------------------------------------------------------------------------------------
                                               8

$$\frac{\left(\left(\sqrt{2} \sqrt{x} - 1\right) \left(- \frac{6}{x^{2}} + \frac{2 \sqrt{2}}{x^{\frac{3}{2}}} + \frac{3 \sqrt{2}}{x^{\frac{5}{2}}}\right) - \frac{6}{x^{2}} + \frac{3 \sqrt{2} \left(\frac{2}{x} - \frac{\sqrt{2}}{x^{\frac{3}{2}}}\right)}{\sqrt{x}} + \frac{3 \sqrt{2}}{x^{\frac{5}{2}}}\right) e^{\sqrt{2 x}}}{8}$$

Simplify

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Derivative of e^sqrt(2x)(sqrt(2*x)-1)

The graph:

Piecewise:

The solution

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How to use it?

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Derivative of e^sqrt(2*x)*(sqrt(2*x)-1)

The graph:

Piecewise:

The solution

Derivative of e^sqrt(2x)(sqrt(2*x)-1)