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Derivative of:
Derivative of e^2*x
Derivative of cos(x)^2
Derivative of x^sin(x)
Derivative of sin(x^3)
Identical expressions
(one +sin(3x))/(one -cos(3x))
(1 plus sinus of (3x)) divide by (1 minus co sinus of e of (3x))
(one plus sinus of (3x)) divide by (one minus co sinus of e of (3x))
1+sin3x/1-cos3x
(1+sin(3x)) divide by (1-cos(3x))
Similar expressions
(1+sin(3x))/(1+cos(3x))
(1-sin(3x))/(1-cos(3x))

The derivative of the function/ (1+sin(3x))/(1-cos(3x))

Derivative of (1+sin(3x))/(1-cos(3x))

The solution

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1 + sin(3*x)
------------
1 - cos(3*x)

$$\frac{\sin{\left(3 x \right)} + 1}{1 - \cos{\left(3 x \right)}}$$

(1 + sin(3*x))/(1 - cos(3*x))

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(3 x \right)} + 1$ and $g{\left(x \right)} = 1 - \cos{\left(3 x \right)}$ .

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

$\frac{3 \left(1 - \cos{\left(3 x \right)}\right) \cos{\left(3 x \right)} - 3 \left(\sin{\left(3 x \right)} + 1\right) \sin{\left(3 x \right)}}{\left(1 - \cos{\left(3 x \right)}\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 3*cos(3*x)    3*(1 + sin(3*x))*sin(3*x)
------------ - -------------------------
1 - cos(3*x)                      2     
                    (1 - cos(3*x))

$$\frac{3 \cos{\left(3 x \right)}}{1 - \cos{\left(3 x \right)}} - \frac{3 \left(\sin{\left(3 x \right)} + 1\right) \sin{\left(3 x \right)}}{\left(1 - \cos{\left(3 x \right)}\right)^{2}}$$

Simplify

The second derivative [src]

  /                 /      2                 \                                 \
  |                 | 2*sin (3*x)            |                                 |
  |  (1 + sin(3*x))*|------------- + cos(3*x)|                                 |
  |                 \-1 + cos(3*x)           /   2*cos(3*x)*sin(3*x)           |
9*|- ----------------------------------------- - ------------------- + sin(3*x)|
  \                -1 + cos(3*x)                    -1 + cos(3*x)              /
--------------------------------------------------------------------------------
                                 -1 + cos(3*x)

$$\frac{9 \left(- \frac{\left(\sin{\left(3 x \right)} + 1\right) \left(\cos{\left(3 x \right)} + \frac{2 \sin^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1}\right)}{\cos{\left(3 x \right)} - 1} + \sin{\left(3 x \right)} - \frac{2 \sin{\left(3 x \right)} \cos{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1}\right)}{\cos{\left(3 x \right)} - 1}$$

Simplify

The third derivative [src]

   /                                                                       /                            2        \                    \
   |                  /      2                 \                           |       6*cos(3*x)      6*sin (3*x)   |                    |
   |                  | 2*sin (3*x)            |            (1 + sin(3*x))*|-1 + ------------- + ----------------|*sin(3*x)           |
   |      2         3*|------------- + cos(3*x)|*cos(3*x)                  |     -1 + cos(3*x)                  2|                    |
   | 3*sin (3*x)      \-1 + cos(3*x)           /                           \                     (-1 + cos(3*x)) /                    |
27*|------------- - ------------------------------------- - --------------------------------------------------------------- + cos(3*x)|
   \-1 + cos(3*x)               -1 + cos(3*x)                                        -1 + cos(3*x)                                    /
---------------------------------------------------------------------------------------------------------------------------------------
                                                             -1 + cos(3*x)

$$\frac{27 \left(- \frac{\left(\sin{\left(3 x \right)} + 1\right) \left(-1 + \frac{6 \cos{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1} + \frac{6 \sin^{2}{\left(3 x \right)}}{\left(\cos{\left(3 x \right)} - 1\right)^{2}}\right) \sin{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1} + \cos{\left(3 x \right)} - \frac{3 \left(\cos{\left(3 x \right)} + \frac{2 \sin^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1}\right) \cos{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1} + \frac{3 \sin^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)} - 1}\right)}{\cos{\left(3 x \right)} - 1}$$

Simplify

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Derivative of (1+sin(3x))/(1-cos(3x))

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