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Identical expressions
sqrt(exp(-x)+8exp(3x^ two))
square root of ( exponent of ( minus x) plus 8 exponent of (3x squared ))
square root of ( exponent of ( minus x) plus 8 exponent of (3x to the power of two))
√(exp(-x)+8exp(3x^2))
sqrt(exp(-x)+8exp(3x2))
sqrtexp-x+8exp3x2
sqrt(exp(-x)+8exp(3x²))
sqrt(exp(-x)+8exp(3x to the power of 2))
sqrtexp-x+8exp3x^2
Similar expressions
sqrt(exp(-x)-8exp(3x^2))
sqrt(exp(x)+8exp(3x^2))

The derivative of the function/ sqrt(exp(-x)+8exp(3x^2))

Derivative of sqrt(exp(-x)+8exp(3x^2))

The solution

You have entered [src]

    _______________
   /             2 
  /   -x      3*x  
\/   e   + 8*e

$$\sqrt{8 e^{3 x^{2}} + e^{- x}}$$

sqrt(exp(-x) + 8*exp(3*x^2))

Detail solution

Let $u = 8 e^{3 x^{2}} + e^{- x}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(8 e^{3 x^{2}} + e^{- x}\right)$ :
1. Differentiate $8 e^{3 x^{2}} + e^{- x}$ term by term:
  1. Let $u = - x$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- x\right)$ :
    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: $-1$
    The result of the chain rule is:
    
    $- e^{- x}$
  4. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = 3 x^{2}$ .
    2. The derivative of $e^{u}$ is itself.
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 3 x^{2}$ :
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        So, the result is: $6 x$
      The result of the chain rule is:
      
      $6 x e^{3 x^{2}}$
    So, the result is: $48 x e^{3 x^{2}}$
  The result is: $48 x e^{3 x^{2}} - e^{- x}$
The result of the chain rule is:

$\frac{48 x e^{3 x^{2}} - e^{- x}}{2 \sqrt{8 e^{3 x^{2}} + e^{- x}}}$
Now simplify:

$\frac{\left(48 x e^{x \left(3 x + 1\right)} - 1\right) e^{- \frac{x}{2}}}{2 \sqrt{8 e^{3 x^{2} + x} + 1}}$

The answer is:

$\frac{\left(48 x e^{x \left(3 x + 1\right)} - 1\right) e^{- \frac{x}{2}}}{2 \sqrt{8 e^{3 x^{2} + x} + 1}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    -x            2
   e           3*x 
 - --- + 24*x*e    
    2              
-------------------
    _______________
   /             2 
  /   -x      3*x  
\/   e   + 8*e

$$\frac{24 x e^{3 x^{2}} - \frac{e^{- x}}{2}}{\sqrt{8 e^{3 x^{2}} + e^{- x}}}$$

Simplify

The second derivative [src]

                                                    2
                                /                 2\ 
 -x          2              2   |   -x         3*x | 
e         3*x         2  3*x    \- e   + 48*x*e    / 
--- + 24*e     + 144*x *e     - ---------------------
 2                                  /      2      \  
                                    |   3*x     -x|  
                                  4*\8*e     + e  /  
-----------------------------------------------------
                     _______________                 
                    /       2                        
                   /     3*x     -x                  
                 \/   8*e     + e

$$\frac{144 x^{2} e^{3 x^{2}} - \frac{\left(48 x e^{3 x^{2}} - e^{- x}\right)^{2}}{4 \left(8 e^{3 x^{2}} + e^{- x}\right)} + 24 e^{3 x^{2}} + \frac{e^{- x}}{2}}{\sqrt{8 e^{3 x^{2}} + e^{- x}}}$$

Simplify

The third derivative [src]

                                                           3                                                         
                                       /                 2\      /                 2\ /       2              2      \
   -x             2              2     |   -x         3*x |      |   -x         3*x | |    3*x         2  3*x     -x|
  e            3*x         3  3*x    3*\- e   + 48*x*e    /    3*\- e   + 48*x*e    /*\48*e     + 288*x *e     + e  /
- --- + 432*x*e     + 864*x *e     + ----------------------- - ------------------------------------------------------
   2                                                     2                         /      2      \                   
                                          /      2      \                          |   3*x     -x|                   
                                          |   3*x     -x|                        4*\8*e     + e  /                   
                                        8*\8*e     + e  /                                                            
---------------------------------------------------------------------------------------------------------------------
                                                     _______________                                                 
                                                    /       2                                                        
                                                   /     3*x     -x                                                  
                                                 \/   8*e     + e

$$\frac{864 x^{3} e^{3 x^{2}} + 432 x e^{3 x^{2}} + \frac{3 \left(48 x e^{3 x^{2}} - e^{- x}\right)^{3}}{8 \left(8 e^{3 x^{2}} + e^{- x}\right)^{2}} - \frac{3 \left(48 x e^{3 x^{2}} - e^{- x}\right) \left(288 x^{2} e^{3 x^{2}} + 48 e^{3 x^{2}} + e^{- x}\right)}{4 \left(8 e^{3 x^{2}} + e^{- x}\right)} - \frac{e^{- x}}{2}}{\sqrt{8 e^{3 x^{2}} + e^{- x}}}$$

Simplify

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Derivative of sqrt(exp(-x)+8exp(3x^2))

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The solution