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Derivative of:
Derivative of x^x^x
Derivative of x^-11
Derivative of -x/2
Derivative of y=3x²+5x+4
Identical expressions
ctg(xe^(-x)+3sqrtx)
ctg(xe to the power of ( minus x) plus 3 square root of x)
ctg(xe^(-x)+3√x)
ctg(xe(-x)+3sqrtx)
ctgxe-x+3sqrtx
ctgxe^-x+3sqrtx
Similar expressions
ctg(xe^(x)+3sqrtx)
ctg(xe^(-x)-3sqrtx)

The derivative of the function/ ctg(xe^(-x)+3sqrtx)

Derivative of ctg(xe^(-x)+3sqrtx)

The solution

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   /   -x       ___\
cot\x*E   + 3*\/ x /

$$\cot{\left(3 \sqrt{x} + e^{- x} x \right)}$$

cot(x*E^(-x) + 3*sqrt(x))

Detail solution

There are multiple ways to do this derivative.
Method #1
1. Rewrite the function to be differentiated:
  
  $\cot{\left(3 \sqrt{x} + e^{- x} x \right)} = \frac{1}{\tan{\left(3 \sqrt{x} + e^{- x} x \right)}}$
2. Let $u = \tan{\left(3 \sqrt{x} + e^{- x} x \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(3 \sqrt{x} + e^{- x} x \right)}$ :
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(3 \sqrt{x} + e^{- x} x \right)} = \frac{\sin{\left(3 \sqrt{x} + e^{- x} x \right)}}{\cos{\left(3 \sqrt{x} + e^{- x} x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(3 \sqrt{x} + e^{- x} x \right)}$ and $g{\left(x \right)} = \cos{\left(3 \sqrt{x} + e^{- x} x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 3 \sqrt{x} + e^{- x} x$ .
    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 $\frac{d}{d x} \left(3 \sqrt{x} + e^{- x} x\right)$ :
      
      Differentiate $3 \sqrt{x} + e^{- x} x$ term by term:
      
      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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      Apply the power rule: $x$ goes to $1$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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 derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      So, the result is: $\frac{3}{2 \sqrt{x}}$
      
      The result is: $\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}$
      
      The result of the chain rule is:
      
      $\left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \cos{\left(3 \sqrt{x} + e^{- x} x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 3 \sqrt{x} + e^{- x} x$ .
    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 $\frac{d}{d x} \left(3 \sqrt{x} + e^{- x} x\right)$ :
      
      Differentiate $3 \sqrt{x} + e^{- x} x$ term by term:
      
      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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      Apply the power rule: $x$ goes to $1$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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 derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      So, the result is: $\frac{3}{2 \sqrt{x}}$
      
      The result is: $\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}$
      
      The result of the chain rule is:
      
      $- \left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \sin{\left(3 \sqrt{x} + e^{- x} x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \sin^{2}{\left(3 \sqrt{x} + e^{- x} x \right)} + \left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \cos^{2}{\left(3 \sqrt{x} + e^{- x} x \right)}}{\cos^{2}{\left(3 \sqrt{x} + e^{- x} x \right)}}$
  The result of the chain rule is:
  
  $- \frac{\left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \sin^{2}{\left(3 \sqrt{x} + e^{- x} x \right)} + \left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \cos^{2}{\left(3 \sqrt{x} + e^{- x} x \right)}}{\cos^{2}{\left(3 \sqrt{x} + e^{- x} x \right)} \tan^{2}{\left(3 \sqrt{x} + e^{- x} x \right)}}$
Method #2
1. Rewrite the function to be differentiated:
  
  $\cot{\left(3 \sqrt{x} + e^{- x} x \right)} = \frac{\cos{\left(3 \sqrt{x} + e^{- x} x \right)}}{\sin{\left(3 \sqrt{x} + e^{- x} x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \cos{\left(3 \sqrt{x} + e^{- x} x \right)}$ and $g{\left(x \right)} = \sin{\left(3 \sqrt{x} + e^{- x} x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Let $u = 3 \sqrt{x} + e^{- x} x$ .
  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 $\frac{d}{d x} \left(3 \sqrt{x} + e^{- x} x\right)$ :
    1. Differentiate $3 \sqrt{x} + e^{- x} x$ term by term:
      
      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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      Apply the power rule: $x$ goes to $1$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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 derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      So, the result is: $\frac{3}{2 \sqrt{x}}$
      
      The result is: $\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}$
    The result of the chain rule is:
    
    $- \left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \sin{\left(3 \sqrt{x} + e^{- x} x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 3 \sqrt{x} + e^{- x} x$ .
  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 $\frac{d}{d x} \left(3 \sqrt{x} + e^{- x} x\right)$ :
    1. Differentiate $3 \sqrt{x} + e^{- x} x$ term by term:
      
      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 $\frac{d}{d x} f{\left(x \right)}$ :
      
      Apply the power rule: $x$ goes to $1$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      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 derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
      
      So, the result is: $\frac{3}{2 \sqrt{x}}$
      
      The result is: $\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}$
    The result of the chain rule is:
    
    $\left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \cos{\left(3 \sqrt{x} + e^{- x} x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- \left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \sin^{2}{\left(3 \sqrt{x} + e^{- x} x \right)} - \left(\left(- x e^{x} + e^{x}\right) e^{- 2 x} + \frac{3}{2 \sqrt{x}}\right) \cos^{2}{\left(3 \sqrt{x} + e^{- x} x \right)}}{\sin^{2}{\left(3 \sqrt{x} + e^{- x} x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/       2/    ___      -x\\ /                                                    2                     \
|1   cot \3*\/ x  + x*e  /| | 3        -x        -x     /   -x     3          -x\     /    ___      -x\|
|- + ---------------------|*|---- + 8*e   - 4*x*e   + 2*|2*e   + ----- - 2*x*e  | *cot\3*\/ x  + x*e  /|
\4             4          / | 3/2                       |          ___          |                      |
                            \x                          \        \/ x           /                      /

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

Simplify

The third derivative [src]

 /       2/    ___      -x\\ /                                                     3                                                          3                                                                                                  \
 |1   cot \3*\/ x  + x*e  /| | 9         -x        -x     /   -x     3          -x\  /       2/    ___      -x\\     /   -x     3          -x\     2/    ___      -x\     /   -x     3          -x\ / 3        -x        -x\    /    ___      -x\|
-|- + ---------------------|*|---- + 24*e   - 8*x*e   + 2*|2*e   + ----- - 2*x*e  | *\1 + cot \3*\/ x  + x*e  // + 4*|2*e   + ----- - 2*x*e  | *cot \3*\/ x  + x*e  / + 6*|2*e   + ----- - 2*x*e  |*|---- + 8*e   - 4*x*e  |*cot\3*\/ x  + x*e  /|
 \8             8          / | 5/2                        |          ___          |                                  |          ___          |                            |          ___          | | 3/2                  |                     |
                             \x                           \        \/ x           /                                  \        \/ x           /                            \        \/ x           / \x                     /                     /

$$- \left(\frac{\cot^{2}{\left(3 \sqrt{x} + x e^{- x} \right)}}{8} + \frac{1}{8}\right) \left(- 8 x e^{- x} + 2 \left(\cot^{2}{\left(3 \sqrt{x} + x e^{- x} \right)} + 1\right) \left(- 2 x e^{- x} + 2 e^{- x} + \frac{3}{\sqrt{x}}\right)^{3} + 6 \left(- 4 x e^{- x} + 8 e^{- x} + \frac{3}{x^{\frac{3}{2}}}\right) \left(- 2 x e^{- x} + 2 e^{- x} + \frac{3}{\sqrt{x}}\right) \cot{\left(3 \sqrt{x} + x e^{- x} \right)} + 4 \left(- 2 x e^{- x} + 2 e^{- x} + \frac{3}{\sqrt{x}}\right)^{3} \cot^{2}{\left(3 \sqrt{x} + x e^{- x} \right)} + 24 e^{- x} + \frac{9}{x^{\frac{5}{2}}}\right)$$

Simplify

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Derivative of ctg(xe^(-x)+3sqrtx)

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Piecewise:

The solution

Method #1

Method #2