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[√tan^- one (cosecx)]
[√ tangent of to the power of minus 1( co sinus of e of ecx)]
[√ tangent of to the power of minus one ( co sinus of e of ecx)]
[√tan-1(cosecx)]
[√tan-1cosecx]
[√tan^-1cosecx]
Similar expressions
[√tan^+1(cosecx)]

The derivative of the function/ [√tan^-1(cosecx)]

Derivative of [√tan^-1(cosecx)]

The solution

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         1         
-------------------
  _________________
\/ tan(cos(E)*c*x)

$$\frac{1}{\sqrt{\tan{\left(x c \cos{\left(e \right)} \right)}}}$$

1/(sqrt(tan((cos(E)*c)*x)))

Detail solution

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

$- \frac{c \sin^{2}{\left(x c \cos{\left(e \right)} \right)} \cos{\left(e \right)} + c \cos{\left(e \right)} \cos^{2}{\left(x c \cos{\left(e \right)} \right)}}{2 \cos^{2}{\left(x c \cos{\left(e \right)} \right)} \tan^{\frac{3}{2}}{\left(x c \cos{\left(e \right)} \right)}}$
Now simplify:

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

The answer is:

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

The first derivative [src]

     /       2            \          
  -c*\1 + tan (cos(E)*c*x)/*cos(E)   
-------------------------------------
                    _________________
2*tan(cos(E)*c*x)*\/ tan(cos(E)*c*x)

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

Simplify

The second derivative [src]

                                  /       /       2            \\
 2    2    /       2            \ |     3*\1 + tan (c*x*cos(E))/|
c *cos (E)*\1 + tan (c*x*cos(E))/*|-1 + ------------------------|
                                  |             2               |
                                  \        4*tan (c*x*cos(E))   /
-----------------------------------------------------------------
                         _________________                       
                       \/ tan(c*x*cos(E))

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

Simplify

The third derivative [src]

                                  /                                                   2                           \
                                  |                             /       2            \      /       2            \|
 3    3    /       2            \ |      _________________   15*\1 + tan (c*x*cos(E))/    7*\1 + tan (c*x*cos(E))/|
c *cos (E)*\1 + tan (c*x*cos(E))/*|- 2*\/ tan(c*x*cos(E))  - -------------------------- + ------------------------|
                                  |                                  7/2                         3/2              |
                                  \                             8*tan   (c*x*cos(E))        2*tan   (c*x*cos(E))  /

$$c^{3} \left(\tan^{2}{\left(c x \cos{\left(e \right)} \right)} + 1\right) \left(- \frac{15 \left(\tan^{2}{\left(c x \cos{\left(e \right)} \right)} + 1\right)^{2}}{8 \tan^{\frac{7}{2}}{\left(c x \cos{\left(e \right)} \right)}} + \frac{7 \left(\tan^{2}{\left(c x \cos{\left(e \right)} \right)} + 1\right)}{2 \tan^{\frac{3}{2}}{\left(c x \cos{\left(e \right)} \right)}} - 2 \sqrt{\tan{\left(c x \cos{\left(e \right)} \right)}}\right) \cos^{3}{\left(e \right)}$$

Simplify

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Derivative of [√tan^-1(cosecx)]

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