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

Derivative of [√tan^-1(cosecx)]

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
         1         
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  _________________
\/ tan(cos(E)*c*x)
1tan⁡(xccos⁡(e))\frac{1}{\sqrt{\tan{\left(x c \cos{\left(e \right)} \right)}}} 
1/(sqrt(tan((cos(E)*c)*x)))
Detail solution

  1. Let u=tan⁡(xccos⁡(e))u = \sqrt{\tan{\left(x c \cos{\left(e \right)} \right)}}.

  2. Apply the power rule: 1u\frac{1}{u} goes to −1u2- \frac{1}{u^{2}}

  3. Then, apply the chain rule. Multiply by ∂∂xtan⁡(xccos⁡(e))\frac{\partial}{\partial x} \sqrt{\tan{\left(x c \cos{\left(e \right)} \right)}}:

    1. Let u=tan⁡(xccos⁡(e))u = \tan{\left(x c \cos{\left(e \right)} \right)}.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ∂∂xtan⁡(xccos⁡(e))\frac{\partial}{\partial x} \tan{\left(x c \cos{\left(e \right)} \right)}:

      1. Rewrite the function to be differentiated:

        tan⁡(xccos⁡(e))=sin⁡(xccos⁡(e))cos⁡(xccos⁡(e))\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:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(xccos⁡(e))f{\left(x \right)} = \sin{\left(x c \cos{\left(e \right)} \right)} and g(x)=cos⁡(xccos⁡(e))g{\left(x \right)} = \cos{\left(x c \cos{\left(e \right)} \right)}.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. Let u=xccos⁡(e)u = x c \cos{\left(e \right)}.

        2. The derivative of sine is cosine:

          ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

        3. Then, apply the chain rule. Multiply by ∂∂xxccos⁡(e)\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.

            1. Apply the power rule: xx goes to 11

            So, the result is: ccos⁡(e)c \cos{\left(e \right)}

          The result of the chain rule is:

          ccos⁡(e)cos⁡(xccos⁡(e))c \cos{\left(e \right)} \cos{\left(x c \cos{\left(e \right)} \right)}

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Let u=xccos⁡(e)u = x c \cos{\left(e \right)}.

        2. The derivative of cosine is negative sine:

          dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

        3. Then, apply the chain rule. Multiply by ∂∂xxccos⁡(e)\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.

            1. Apply the power rule: xx goes to 11

            So, the result is: ccos⁡(e)c \cos{\left(e \right)}

          The result of the chain rule is:

          −csin⁡(xccos⁡(e))cos⁡(e)- c \sin{\left(x c \cos{\left(e \right)} \right)} \cos{\left(e \right)}

        Now plug in to the quotient rule:

        csin⁡2(xccos⁡(e))cos⁡(e)+ccos⁡(e)cos⁡2(xccos⁡(e))cos⁡2(xccos⁡(e))\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:

      csin⁡2(xccos⁡(e))cos⁡(e)+ccos⁡(e)cos⁡2(xccos⁡(e))2cos⁡2(xccos⁡(e))tan⁡(xccos⁡(e))\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:

    −csin⁡2(xccos⁡(e))cos⁡(e)+ccos⁡(e)cos⁡2(xccos⁡(e))2cos⁡2(xccos⁡(e))tan⁡32(xccos⁡(e))- \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)}}

  4. Now simplify:

    −ccos⁡(e)2cos⁡2(cxcos⁡(e))tan⁡32(cxcos⁡(e))- \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:

−ccos⁡(e)2cos⁡2(cxcos⁡(e))tan⁡32(cxcos⁡(e))- \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)
−c(tan⁡2(xccos⁡(e))+1)cos⁡(e)2tan⁡(xccos⁡(e))tan⁡(xccos⁡(e))- \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))
c2(3(tan⁡2(cxcos⁡(e))+1)4tan⁡2(cxcos⁡(e))−1)(tan⁡2(cxcos⁡(e))+1)cos⁡2(e)tan⁡(cxcos⁡(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))  /
c3(tan⁡2(cxcos⁡(e))+1)(−15(tan⁡2(cxcos⁡(e))+1)28tan⁡72(cxcos⁡(e))+7(tan⁡2(cxcos⁡(e))+1)2tan⁡32(cxcos⁡(e))−2tan⁡(cxcos⁡(e)))cos⁡3(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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