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Identical expressions
(x/sin(3x)^ two)- twenty-six
(x divide by sinus of (3x) squared ) minus 26
(x divide by sinus of (3x) to the power of two) minus twenty minus six
(x/sin(3x)2)-26
x/sin3x2-26
(x/sin(3x)²)-26
(x/sin(3x) to the power of 2)-26
x/sin3x^2-26
(x divide by sin(3x)^2)-26
Similar expressions
(x/sin(3x)^2)+26

The derivative of the function/ (x/sin(3x)^2)-26

Derivative of (x/sin(3x)^2)-26

The solution

You have entered [src]

    x         
--------- - 26
   2          
sin (3*x)

$$\frac{x}{\sin^{2}{\left(3 x \right)}} - 26$$

d /    x         \
--|--------- - 26|
dx|   2          |
  \sin (3*x)     /

$$\frac{d}{d x} \left(\frac{x}{\sin^{2}{\left(3 x \right)}} - 26\right)$$

Transform

Detail solution

Differentiate $\frac{x}{\sin^{2}{\left(3 x \right)}} - 26$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = x$ and $g{\left(x \right)} = \sin^{2}{\left(3 x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \sin{\left(3 x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
    1. Let $u = 3 x$ .
    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 $\frac{d}{d x} 3 x$ :
      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:
      
      $3 \cos{\left(3 x \right)}$
    The result of the chain rule is:
    
    $6 \sin{\left(3 x \right)} \cos{\left(3 x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- 6 x \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \sin^{2}{\left(3 x \right)}}{\sin^{4}{\left(3 x \right)}}$
2. The derivative of the constant $\left(-1\right) 26$ is zero.
The result is: $\frac{- 6 x \sin{\left(3 x \right)} \cos{\left(3 x \right)} + \sin^{2}{\left(3 x \right)}}{\sin^{4}{\left(3 x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1       6*x*cos(3*x)
--------- - ------------
   2            3       
sin (3*x)    sin (3*x)

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

Simplify

The second derivative [src]

  /                          2     \
  |      2*cos(3*x)   9*x*cos (3*x)|
6*|3*x - ---------- + -------------|
  |       sin(3*x)         2       |
  \                     sin (3*x)  /
------------------------------------
                2                   
             sin (3*x)

$$\frac{6 \cdot \left(3 x + \frac{9 x \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}} - \frac{2 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}\right)}{\sin^{2}{\left(3 x \right)}}$$

Simplify

The third derivative [src]

   /         2                3                    \
   |    3*cos (3*x)   12*x*cos (3*x)   8*x*cos(3*x)|
54*|1 + ----------- - -------------- - ------------|
   |        2              3             sin(3*x)  |
   \     sin (3*x)      sin (3*x)                  /
----------------------------------------------------
                        2                           
                     sin (3*x)

$$\frac{54 \left(- \frac{8 x \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}} - \frac{12 x \cos^{3}{\left(3 x \right)}}{\sin^{3}{\left(3 x \right)}} + 1 + \frac{3 \cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right)}{\sin^{2}{\left(3 x \right)}}$$

Simplify

The graph

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Derivative of (x/sin(3x)^2)-26

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The solution