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Derivative of sin(x)/(1+cos(x))
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Derivative of 8*x
Identical expressions
(sqrt(tan(x/(e^x))))
( square root of ( tangent of (x divide by (e to the power of x))))
(√(tan(x/(e^x))))
(sqrt(tan(x/(ex))))
sqrttanx/ex
sqrttanx/e^x
(sqrt(tan(x divide by (e^x))))

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

Derivative of (sqrt(tan(x/(e^x))))

The solution

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     _________
    /    /x \ 
   /  tan|--| 
  /      | x| 
\/       \E /

$$\sqrt{\tan{\left(\frac{x}{e^{x}} \right)}}$$

sqrt(tan(x/E^x))

Detail solution

Let $u = \tan{\left(\frac{x}{e^{x}} \right)}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(\frac{x}{e^{x}} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\tan{\left(\frac{x}{e^{x}} \right)} = \frac{\sin{\left(\frac{x}{e^{x}} \right)}}{\cos{\left(\frac{x}{e^{x}} \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \sin{\left(\frac{x}{e^{x}} \right)}$ and $g{\left(x \right)} = \cos{\left(\frac{x}{e^{x}} \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = \frac{x}{e^{x}}$ .
  2. The derivative of sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{e^{x}}$ :
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = x$ and $g{\left(x \right)} = e^{x}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of $e^{x}$ is itself.
      Now plug in to the quotient rule:
      
      $\left(- x e^{x} + e^{x}\right) e^{- 2 x}$
    The result of the chain rule is:
    
    $\left(- x e^{x} + e^{x}\right) e^{- 2 x} \cos{\left(\frac{x}{e^{x}} \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \frac{x}{e^{x}}$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{e^{x}}$ :
    1. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = x$ and $g{\left(x \right)} = e^{x}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Apply the power rule: $x$ goes to $1$
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of $e^{x}$ is itself.
      Now plug in to the quotient rule:
      
      $\left(- x e^{x} + e^{x}\right) e^{- 2 x}$
    The result of the chain rule is:
    
    $- \left(- x e^{x} + e^{x}\right) e^{- 2 x} \sin{\left(\frac{x}{e^{x}} \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\left(- x e^{x} + e^{x}\right) e^{- 2 x} \sin^{2}{\left(\frac{x}{e^{x}} \right)} + \left(- x e^{x} + e^{x}\right) e^{- 2 x} \cos^{2}{\left(\frac{x}{e^{x}} \right)}}{\cos^{2}{\left(\frac{x}{e^{x}} \right)}}$
The result of the chain rule is:

$\frac{\left(- x e^{x} + e^{x}\right) e^{- 2 x} \sin^{2}{\left(\frac{x}{e^{x}} \right)} + \left(- x e^{x} + e^{x}\right) e^{- 2 x} \cos^{2}{\left(\frac{x}{e^{x}} \right)}}{2 \cos^{2}{\left(\frac{x}{e^{x}} \right)} \sqrt{\tan{\left(\frac{x}{e^{x}} \right)}}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

/       2/x \\ /1       -x\
|1 + tan |--||*|-- - x*e  |
|        | x|| | x        |
\        \E // \E         /
---------------------------
             _________     
            /    /x \      
      2*   /  tan|--|      
          /      | x|      
        \/       \E /

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

Simplify

The second derivative [src]

                  /                                 ____________               2 /       2/   -x\\  -x\    
/       2/   -x\\ |      -2 + x                2   /    /   -x\   -x   (-1 + x) *\1 + tan \x*e  //*e  |  -x
\1 + tan \x*e  //*|----------------- + (-1 + x) *\/  tan\x*e  / *e   - -------------------------------|*e  
                  |     ____________                                                3/2/   -x\        |    
                  |    /    /   -x\                                            4*tan   \x*e  /        |    
                  \2*\/  tan\x*e  /                                                                   /

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

Simplify

The third derivative [src]

                  /                                                                                                                                                         2                                                            \    
                  |                                                                3 /       2/   -x\\  -2*x        ____________                           /       2/   -x\\          3  -2*x     /       2/   -x\\                    -x|    
/       2/   -x\\ |        -3 + x                  3    3/2/   -x\  -2*x   (-1 + x) *\1 + tan \x*e  //*e           /    /   -x\                     -x   3*\1 + tan \x*e  // *(-1 + x) *e       3*\1 + tan \x*e  //*(-1 + x)*(-2 + x)*e  |  -x
\1 + tan \x*e  //*|- ----------------- - 2*(-1 + x) *tan   \x*e  /*e     + --------------------------------- - 3*\/  tan\x*e  / *(-1 + x)*(-2 + x)*e   - ------------------------------------ + -----------------------------------------|*e  
                  |       ____________                                                  ____________                                                                    5/2/   -x\                                3/2/   -x\             |    
                  |      /    /   -x\                                                  /    /   -x\                                                                8*tan   \x*e  /                           4*tan   \x*e  /             |    
                  \  2*\/  tan\x*e  /                                              2*\/  tan\x*e  /                                                                                                                                      /

$$\left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) \left(- \frac{x - 3}{2 \sqrt{\tan{\left(x e^{- x} \right)}}} + \frac{3 \left(x - 2\right) \left(x - 1\right) \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) e^{- x}}{4 \tan^{\frac{3}{2}}{\left(x e^{- x} \right)}} - 3 \left(x - 2\right) \left(x - 1\right) e^{- x} \sqrt{\tan{\left(x e^{- x} \right)}} - \frac{3 \left(x - 1\right)^{3} \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right)^{2} e^{- 2 x}}{8 \tan^{\frac{5}{2}}{\left(x e^{- x} \right)}} + \frac{\left(x - 1\right)^{3} \left(\tan^{2}{\left(x e^{- x} \right)} + 1\right) e^{- 2 x}}{2 \sqrt{\tan{\left(x e^{- x} \right)}}} - 2 \left(x - 1\right)^{3} e^{- 2 x} \tan^{\frac{3}{2}}{\left(x e^{- x} \right)}\right) e^{- x}$$

Simplify

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Identical expressions

Derivative of (sqrt(tan(x/(e^x))))

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The solution