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(1+cscx)/(1-cscx)
The derivative of the function/ (1+cscx)/(1-cscx)

Derivative of (1+cscx)/(1-cscx)

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

You have entered [src] 
1 + csc(x)
----------
1 - csc(x)
csc⁡(x)+1−csc⁡(x)+1\frac{\csc{\left(x \right)} + 1}{- \csc{\left(x \right)} + 1} 
d /1 + csc(x)\
--|----------|
dx\1 - csc(x)/
ddxcsc⁡(x)+1−csc⁡(x)+1\frac{d}{d x} \frac{\csc{\left(x \right)} + 1}{- \csc{\left(x \right)} + 1}
Transform
Detail solution

  1. 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)=csc⁡(x)+1f{\left(x \right)} = \csc{\left(x \right)} + 1 and g(x)=1−csc⁡(x)g{\left(x \right)} = 1 - \csc{\left(x \right)}.

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

    1. Differentiate csc⁡(x)+1\csc{\left(x \right)} + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Rewrite the function to be differentiated:

        csc⁡(x)=1sin⁡(x)\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}

      3. Let u=sin⁡(x)u = \sin{\left(x \right)}.

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

      5. Then, apply the chain rule. Multiply by ddxsin⁡(x)\frac{d}{d x} \sin{\left(x \right)}:

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        The result of the chain rule is:

        −cos⁡(x)sin⁡2(x)- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

      The result is: −cos⁡(x)sin⁡2(x)- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

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

    1. Differentiate 1−csc⁡(x)1 - \csc{\left(x \right)} term by term:

      1. The derivative of the constant 11 is zero.

      2. The derivative of a constant times a function is the constant times the derivative of the function.

        1. The derivative of cosecant is negative cosecant times cotangent:

          ddxcsc⁡(x)=−cot⁡(x)csc⁡(x)\frac{d}{d x} \csc{\left(x \right)} = - \cot{\left(x \right)} \csc{\left(x \right)}

        So, the result is: cos⁡(x)sin⁡2(x)\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

      The result is: cos⁡(x)sin⁡2(x)\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}

    Now plug in to the quotient rule:

    −(1−csc⁡(x))cos⁡(x)sin⁡2(x)−(csc⁡(x)+1)cos⁡(x)sin⁡2(x)(1−csc⁡(x))2\frac{- \frac{\left(1 - \csc{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(\csc{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}}{\left(1 - \csc{\left(x \right)}\right)^{2}}

  2. Now simplify:

    −2cos⁡(x)(sin⁡(x)−1)2- \frac{2 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}

The answer is:

−2cos⁡(x)(sin⁡(x)−1)2- \frac{2 \cos{\left(x \right)}}{\left(\sin{\left(x \right)} - 1\right)^{2}}

The graph
02468-8-6-4-2-1010-200000200000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  cot(x)*csc(x)   (1 + csc(x))*cot(x)*csc(x)
- ------------- - --------------------------
    1 - csc(x)                      2       
                        (1 - csc(x))
−cot⁡(x)csc⁡(x)−csc⁡(x)+1−(csc⁡(x)+1)cot⁡(x)csc⁡(x)(−csc⁡(x)+1)2- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{- \csc{\left(x \right)} + 1} - \frac{\left(\csc{\left(x \right)} + 1\right) \cot{\left(x \right)} \csc{\left(x \right)}}{\left(- \csc{\left(x \right)} + 1\right)^{2}}
Simplify
The second derivative [src] 
/                              /                     2          \                   \       
|                              |         2      2*cot (x)*csc(x)|                   |       
|                 (1 + csc(x))*|1 + 2*cot (x) - ----------------|        2          |       
|          2                   \                  -1 + csc(x)   /   2*cot (x)*csc(x)|       
|-1 - 2*cot (x) + ----------------------------------------------- + ----------------|*csc(x)
\                                   -1 + csc(x)                       -1 + csc(x)   /       
--------------------------------------------------------------------------------------------
                                        -1 + csc(x)
(−2cot⁡2(x)+2cot⁡2(x)csc⁡(x)csc⁡(x)−1+(csc⁡(x)+1)(2cot⁡2(x)−2cot⁡2(x)csc⁡(x)csc⁡(x)−1+1)csc⁡(x)−1−1)csc⁡(x)csc⁡(x)−1\frac{\left(- 2 \cot^{2}{\left(x \right)} + \frac{2 \cot^{2}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + \frac{\left(\csc{\left(x \right)} + 1\right) \left(2 \cot^{2}{\left(x \right)} - \frac{2 \cot^{2}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 1\right)}{\csc{\left(x \right)} - 1} - 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1}
Simplify
The third derivative [src] 
/                             /                     2               /       2   \               2       2   \                                                                         \              
|                             |         2      6*cot (x)*csc(x)   6*\1 + cot (x)/*csc(x)   6*cot (x)*csc (x)|                                /                     2          \       |              
|                (1 + csc(x))*|5 + 6*cot (x) - ---------------- - ---------------------- + -----------------|                                |         2      2*cot (x)*csc(x)|       |              
|                             |                  -1 + csc(x)           -1 + csc(x)                        2 |     /         2   \          3*|1 + 2*cot (x) - ----------------|*csc(x)|              
|         2                   \                                                              (-1 + csc(x))  /   3*\1 + 2*cot (x)/*csc(x)     \                  -1 + csc(x)   /       |              
|5 + 6*cot (x) - -------------------------------------------------------------------------------------------- - ------------------------ - -------------------------------------------|*cot(x)*csc(x)
\                                                        -1 + csc(x)                                                  -1 + csc(x)                          -1 + csc(x)                /              
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                             -1 + csc(x)
(6cot⁡2(x)−3⋅(2cot⁡2(x)+1)csc⁡(x)csc⁡(x)−1−(csc⁡(x)+1)(6cot⁡2(x)−6cot⁡2(x)csc⁡(x)csc⁡(x)−1+6cot⁡2(x)csc⁡2(x)(csc⁡(x)−1)2−6(cot⁡2(x)+1)csc⁡(x)csc⁡(x)−1+5)csc⁡(x)−1−3⋅(2cot⁡2(x)−2cot⁡2(x)csc⁡(x)csc⁡(x)−1+1)csc⁡(x)csc⁡(x)−1+5)cot⁡(x)csc⁡(x)csc⁡(x)−1\frac{\left(6 \cot^{2}{\left(x \right)} - \frac{3 \cdot \left(2 \cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} - \frac{\left(\csc{\left(x \right)} + 1\right) \left(6 \cot^{2}{\left(x \right)} - \frac{6 \cot^{2}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + \frac{6 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}}{\left(\csc{\left(x \right)} - 1\right)^{2}} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 5\right)}{\csc{\left(x \right)} - 1} - \frac{3 \cdot \left(2 \cot^{2}{\left(x \right)} - \frac{2 \cot^{2}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 5\right) \cot{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1}
Simplify
The graph
Derivative of (1+cscx)/(1-cscx)
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