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Derivative of:
Derivative of x^5*sin(x)
Derivative of (x+5)*sqrt(x)
Derivative of (x^2-x)/2
Derivative of x^(-1/3)
Identical expressions
csc^ five (ln(tanx^2e^x))
csc to the power of 5(ln( tangent of x squared e to the power of x))
csc to the power of five (ln( tangent of x squared e to the power of x))
csc5(ln(tanx2ex))
csc5lntanx2ex
csc⁵(ln(tanx²e^x))
csc to the power of 5(ln(tanx to the power of 2e to the power of x))
csc^5lntanx^2e^x

The derivative of the function/ csc^5(ln(tanx^2e^x))

Derivative of csc^5(ln(tanx^2e^x))

The solution

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   5/   /   2     x\\
csc \log\tan (x)*E //

$$\csc^{5}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}$$

csc(log(tan(x)^2*E^x))^5

Detail solution

Let $u = \csc{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}$ .
Apply the power rule: $u^{5}$ goes to $5 u^{4}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \csc{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}$ :
1. Rewrite the function to be differentiated:
  
  $\csc{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} = \frac{1}{\sin{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}$
2. Let $u = \sin{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}$ .
3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}$ :
  1. Let $u = \log{\left(e^{x} \tan^{2}{\left(x \right)} \right)}$ .
  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 $\frac{d}{d x} \log{\left(e^{x} \tan^{2}{\left(x \right)} \right)}$ :
    1. Let $u = e^{x} \tan^{2}{\left(x \right)}$ .
    2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} e^{x} \tan^{2}{\left(x \right)}$ :
      1. Apply the product rule:
        
        $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
        
        $f{\left(x \right)} = \tan^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
        
        Let $u = \tan{\left(x \right)}$ .
        
        Apply the power rule: $u^{2}$ goes to $2 u$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(x \right)}$ :
        
        Rewrite the function to be differentiated:
        
        $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
        
        Apply the quotient rule, which is:
        
        $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
        
        $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
        
        To find $\frac{d}{d x} f{\left(x \right)}$ :
        
        The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
        
        To find $\frac{d}{d x} g{\left(x \right)}$ :
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
        
        Now plug in to the quotient rule:
        
        $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
        
        The result of the chain rule is:
        
        $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
        
        $g{\left(x \right)} = e^{x}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
        
        The derivative of $e^{x}$ is itself.
        
        The result is: $\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \tan^{2}{\left(x \right)}$
      The result of the chain rule is:
      
      $\frac{\left(\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \tan^{2}{\left(x \right)}\right) e^{- x}}{\tan^{2}{\left(x \right)}}$
    The result of the chain rule is:
    
    $\frac{\left(\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \tan^{2}{\left(x \right)}\right) e^{- x} \cos{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}}$
  The result of the chain rule is:
  
  $- \frac{\left(\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \tan^{2}{\left(x \right)}\right) e^{- x} \cos{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\sin^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} \tan^{2}{\left(x \right)}}$
The result of the chain rule is:

$- \frac{5 \left(\frac{2 \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) e^{x} \tan{\left(x \right)}}{\cos^{2}{\left(x \right)}} + e^{x} \tan^{2}{\left(x \right)}\right) e^{- x} \cos{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} \csc^{4}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\sin^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} \tan^{2}{\left(x \right)}}$
Now simplify:

$- \frac{5 \left(1 + \frac{2}{\sin{\left(x \right)} \cos{\left(x \right)}}\right) \cos{\left(x + \log{\left(\tan^{2}{\left(x \right)} \right)} \right)}}{\sin^{6}{\left(x + \log{\left(\tan^{2}{\left(x \right)} \right)} \right)}}$

The answer is:

$- \frac{5 \left(1 + \frac{2}{\sin{\left(x \right)} \cos{\left(x \right)}}\right) \cos{\left(x + \log{\left(\tan^{2}{\left(x \right)} \right)} \right)}}{\sin^{6}{\left(x + \log{\left(\tan^{2}{\left(x \right)} \right)} \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      5/   /   2     x\\ /   2     x   /         2   \  x       \    /   /   2     x\\  -x
-5*csc \log\tan (x)*E //*\tan (x)*e  + \2 + 2*tan (x)/*e *tan(x)/*cot\log\tan (x)*E //*e  
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                                            2                                             
                                         tan (x)

$$- \frac{5 \left(\left(2 \tan^{2}{\left(x \right)} + 2\right) e^{x} \tan{\left(x \right)} + e^{x} \tan^{2}{\left(x \right)}\right) e^{- x} \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} \csc^{5}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}}$$

Simplify

The second derivative [src]

                        /                                                                        2                               /                         2                                                   \                                                  2                                                                                      \
                        |                                                /         2            \  /       2/   /   2     x\\\   |   2        /       2   \         2    /       2   \     /       2   \       |    /   /   2     x\\     /         2            \     2/   /   2     x\\     /       2   \ /         2            \    /   /   2     x\\|
     5/   /   2     x\\ |/         2            \    /   /   2     x\\   \2 + 2*tan (x) + tan(x)/ *\1 + cot \log\tan (x)*e ///   \tan (x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 4*\1 + tan (x)/*tan(x)/*cot\log\tan (x)*e //   5*\2 + 2*tan (x) + tan(x)/ *cot \log\tan (x)*e //   2*\1 + tan (x)/*\2 + 2*tan (x) + tan(x)/*cot\log\tan (x)*e //|
5*csc \log\tan (x)*e //*|\2 + 2*tan (x) + tan(x)/*cot\log\tan (x)*e // + ----------------------------------------------------- - ---------------------------------------------------------------------------------------------------- + ------------------------------------------------- + -------------------------------------------------------------|
                        \                                                                        tan(x)                                                                         tan(x)                                                                        tan(x)                                                    tan(x)                           /
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                                                                                                                                                                          tan(x)

$$\frac{5 \left(\frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}} + \frac{\left(\cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{2}}{\tan{\left(x \right)}} + \frac{5 \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{2} \cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}} + \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} - \frac{\left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)}\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}}\right) \csc^{5}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}}$$

Simplify

The third derivative [src]

                        /                                                  /                         2                                                                                                 2       \                                                   3                                                    2                                                   2                                 /                         2                                                   \                                                                                                                   2                                                                  3                                                                   2                                                                           2                                                                                                                                                                    /                         2                                                   \                   /                         2                                                   \                                                                          /                         2                                                   \\
                        |                                                  |   2        /       2   \      /       2   \               3    /       2   \         2    /       2   \      /       2   \        |    /   /   2     x\\      /         2            \     3/   /   2     x\\      /         2            \     2/   /   2     x\\     /         2            \  /       2/   /   2     x\\\     |   2        /       2   \         2    /       2   \     /       2   \       |    /   /   2     x\\                                                                      /         2            \     2/   /   2     x\\ /       2   \      /         2            \  /       2/   /   2     x\\\    /   /   2     x\\     /       2   \  /         2            \    /   /   2     x\\     /         2            \  /       2/   /   2     x\\\ /       2   \     /       2   \ /         2            \    /   /   2     x\\     /       2/   /   2     x\\\ /         2            \ |   2        /       2   \         2    /       2   \     /       2   \       |     /       2   \ |   2        /       2   \         2    /       2   \     /       2   \       |    /   /   2     x\\         2/   /   2     x\\ /         2            \ |   2        /       2   \         2    /       2   \     /       2   \       ||
     5/   /   2     x\\ |  /         2            \    /   /   2     x\\   \tan (x) + 6*\1 + tan (x)/  + 6*\1 + tan (x)/*tan(x) + 8*tan (x)*\1 + tan (x)/ + 12*tan (x)*\1 + tan (x)/ + 16*\1 + tan (x)/ *tan(x)/*cot\log\tan (x)*e //   25*\2 + 2*tan (x) + tan(x)/ *cot \log\tan (x)*e //   15*\2 + 2*tan (x) + tan(x)/ *cot \log\tan (x)*e //   3*\2 + 2*tan (x) + tan(x)/ *\1 + cot \log\tan (x)*e ///   2*\tan (x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 4*\1 + tan (x)/*tan(x)/*cot\log\tan (x)*e //     /       2   \ /         2            \    /   /   2     x\\   30*\2 + 2*tan (x) + tan(x)/ *cot \log\tan (x)*e //*\1 + tan (x)/   17*\2 + 2*tan (x) + tan(x)/ *\1 + cot \log\tan (x)*e ///*cot\log\tan (x)*e //   6*\1 + tan (x)/ *\2 + 2*tan (x) + tan(x)/*cot\log\tan (x)*e //   6*\2 + 2*tan (x) + tan(x)/ *\1 + cot \log\tan (x)*e ///*\1 + tan (x)/   4*\1 + tan (x)/*\2 + 2*tan (x) + tan(x)/*cot\log\tan (x)*e //   3*\1 + cot \log\tan (x)*e ///*\2 + 2*tan (x) + tan(x)/*\tan (x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 4*\1 + tan (x)/*tan(x)/   4*\1 + tan (x)/*\tan (x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 4*\1 + tan (x)/*tan(x)/*cot\log\tan (x)*e //   15*cot \log\tan (x)*e //*\2 + 2*tan (x) + tan(x)/*\tan (x) + 2*\1 + tan (x)/  + 4*tan (x)*\1 + tan (x)/ + 4*\1 + tan (x)/*tan(x)/|
5*csc \log\tan (x)*e //*|- \2 + 2*tan (x) + tan(x)/*cot\log\tan (x)*e // - ---------------------------------------------------------------------------------------------------------------------------------------------------------- - -------------------------------------------------- - -------------------------------------------------- - ------------------------------------------------------- + ------------------------------------------------------------------------------------------------------ + 4*\1 + tan (x)/*\2 + 2*tan (x) + tan(x)/*cot\log\tan (x)*e // - ---------------------------------------------------------------- - ----------------------------------------------------------------------------- - -------------------------------------------------------------- - --------------------------------------------------------------------- - ------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------------------------- + -------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------|
                        |                                                                                                                            tan(x)                                                                                                     2                                                  tan(x)                                                  tan(x)                                                                           tan(x)                                                                                                                                                  2                                                                         2                                                                       2                                                                    2                                                                 tan(x)                                                                                                2                                                                                                                               2                                                                                                                             2                                                                |
                        \                                                                                                                                                                                                                                    tan (x)                                                                                                                                                                                                                                                                                                                                             tan (x)                                                                   tan (x)                                                                 tan (x)                                                              tan (x)                                                                                                                                                                 tan (x)                                                                                                                         tan (x)                                                                                                                       tan (x)                                                             /
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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       tan(x)

$$\frac{5 \left(- \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}} - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{2}}{\tan^{2}{\left(x \right)}} - \frac{30 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{2} \cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} - \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)}\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}} - \frac{17 \left(\cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{3} \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}} - \frac{3 \left(\cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{2}}{\tan{\left(x \right)}} + \frac{3 \left(\cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} + 1\right) \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)}\right)}{\tan^{2}{\left(x \right)}} - \frac{25 \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{3} \cot^{3}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}} - \frac{15 \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right)^{2} \cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}} + \frac{15 \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)}\right) \cot^{2}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan^{2}{\left(x \right)}} - \left(2 \tan^{2}{\left(x \right)} + \tan{\left(x \right)} + 2\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)} + \frac{2 \left(2 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)}\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}} - \frac{\left(16 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \tan{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right)^{2} + 8 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{3}{\left(x \right)} + 12 \left(\tan^{2}{\left(x \right)} + 1\right) \tan^{2}{\left(x \right)} + 6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)}\right) \cot{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}}\right) \csc^{5}{\left(\log{\left(e^{x} \tan^{2}{\left(x \right)} \right)} \right)}}{\tan{\left(x \right)}}$$

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How to use it?

Identical expressions

Derivative of csc^5(ln(tanx^2e^x))

The graph:

Piecewise:

The solution