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

The solution

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1 + csc(x)
----------
1 - csc(x)

$$\frac{\csc{\left(x \right)} + 1}{- \csc{\left(x \right)} + 1}$$

d /1 + csc(x)\
--|----------|
dx\1 - csc(x)/

$$\frac{d}{d x} \frac{\csc{\left(x \right)} + 1}{- \csc{\left(x \right)} + 1}$$

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\csc{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Rewrite the function to be differentiated:
    
    $\csc{\left(x \right)} = \frac{1}{\sin{\left(x \right)}}$
  3. Let $u = \sin{\left(x \right)}$ .
  4. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  5. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  The result is: $- \frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $1 - \csc{\left(x \right)}$ term by term:
  1. The derivative of the constant $1$ 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:
      
      $\frac{d}{d x} \csc{\left(x \right)} = - \cot{\left(x \right)} \csc{\left(x \right)}$
    So, the result is: $\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
  The result is: $\frac{\cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}$
Now plug in to the quotient rule:

$\frac{- \frac{\left(1 - \csc{\left(x \right)}\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{\left(\csc{\left(x \right)} + 1\right) \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}}}{\left(1 - \csc{\left(x \right)}\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

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))

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

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)

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

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)

$$\frac{\left(6 \cot^{2}{\left(x \right)} - \frac{3 \cdot \left(2 \cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} - \frac{\left(\csc{\left(x \right)} + 1\right) \left(6 \cot^{2}{\left(x \right)} - \frac{6 \cot^{2}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + \frac{6 \cot^{2}{\left(x \right)} \csc^{2}{\left(x \right)}}{\left(\csc{\left(x \right)} - 1\right)^{2}} - \frac{6 \left(\cot^{2}{\left(x \right)} + 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 5\right)}{\csc{\left(x \right)} - 1} - \frac{3 \cdot \left(2 \cot^{2}{\left(x \right)} - \frac{2 \cot^{2}{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 1\right) \csc{\left(x \right)}}{\csc{\left(x \right)} - 1} + 5\right) \cot{\left(x \right)} \csc{\left(x \right)}}{\csc{\left(x \right)} - 1}$$

Simplify

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

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