Derivative of cos^5x/sin9x

The solution

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   5    
cos (x) 
--------
sin(9*x)

$$\frac{\cos^{5}{\left(x \right)}}{\sin{\left(9 x \right)}}$$

cos(x)^5/sin(9*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos^{5}{\left(x \right)}$ and $g{\left(x \right)} = \sin{\left(9 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  The result of the chain rule is:
  
  $- 5 \sin{\left(x \right)} \cos^{4}{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 9 x$ .
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 $\frac{d}{d x} 9 x$ :
  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: $9$
  The result of the chain rule is:
  
  $9 \cos{\left(9 x \right)}$
Now plug in to the quotient rule:

$\frac{- 5 \sin{\left(x \right)} \sin{\left(9 x \right)} \cos^{4}{\left(x \right)} - 9 \cos^{5}{\left(x \right)} \cos{\left(9 x \right)}}{\sin^{2}{\left(9 x \right)}}$
Now simplify:

$- \frac{\left(7 \cos{\left(8 x \right)} + 2 \cos{\left(10 x \right)}\right) \cos^{4}{\left(x \right)}}{\sin^{2}{\left(9 x \right)}}$

The answer is:

$- \frac{\left(7 \cos{\left(8 x \right)} + 2 \cos{\left(10 x \right)}\right) \cos^{4}{\left(x \right)}}{\sin^{2}{\left(9 x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       5                    4          
  9*cos (x)*cos(9*x)   5*cos (x)*sin(x)
- ------------------ - ----------------
         2                 sin(9*x)    
      sin (9*x)

$$- \frac{5 \sin{\left(x \right)} \cos^{4}{\left(x \right)}}{\sin{\left(9 x \right)}} - \frac{9 \cos^{5}{\left(x \right)} \cos{\left(9 x \right)}}{\sin^{2}{\left(9 x \right)}}$$

Simplify

The second derivative [src]

        /                                      /         2     \                            \
   3    |       2            2            2    |    2*cos (9*x)|   90*cos(x)*cos(9*x)*sin(x)|
cos (x)*|- 5*cos (x) + 20*sin (x) + 81*cos (x)*|1 + -----------| + -------------------------|
        |                                      |        2      |            sin(9*x)        |
        \                                      \     sin (9*x) /                            /
---------------------------------------------------------------------------------------------
                                           sin(9*x)

$$\frac{\left(81 \left(1 + \frac{2 \cos^{2}{\left(9 x \right)}}{\sin^{2}{\left(9 x \right)}}\right) \cos^{2}{\left(x \right)} + 20 \sin^{2}{\left(x \right)} + \frac{90 \sin{\left(x \right)} \cos{\left(x \right)} \cos{\left(9 x \right)}}{\sin{\left(9 x \right)}} - 5 \cos^{2}{\left(x \right)}\right) \cos^{3}{\left(x \right)}}{\sin{\left(9 x \right)}}$$

Simplify

The third derivative [src]

         /                                                                                                                                         /         2     \         \ 
         |                                                                                                                                    3    |    6*cos (9*x)|         | 
         |                                                                                                                             729*cos (x)*|5 + -----------|*cos(9*x)| 
         |                                                    /         2     \              /     2           2   \                               |        2      |         | 
    2    |  /        2            2   \                  2    |    2*cos (9*x)|          135*\- cos (x) + 4*sin (x)/*cos(x)*cos(9*x)               \     sin (9*x) /         | 
-cos (x)*|5*\- 13*cos (x) + 12*sin (x)/*sin(x) + 1215*cos (x)*|1 + -----------|*sin(x) + ------------------------------------------- + --------------------------------------| 
         |                                                    |        2      |                            sin(9*x)                                   sin(9*x)               | 
         \                                                    \     sin (9*x) /                                                                                              / 
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                    sin(9*x)

$$- \frac{\left(1215 \left(1 + \frac{2 \cos^{2}{\left(9 x \right)}}{\sin^{2}{\left(9 x \right)}}\right) \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \frac{729 \left(5 + \frac{6 \cos^{2}{\left(9 x \right)}}{\sin^{2}{\left(9 x \right)}}\right) \cos^{3}{\left(x \right)} \cos{\left(9 x \right)}}{\sin{\left(9 x \right)}} + \frac{135 \left(4 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} \cos{\left(9 x \right)}}{\sin{\left(9 x \right)}} + 5 \left(12 \sin^{2}{\left(x \right)} - 13 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin{\left(9 x \right)}}$$

Simplify

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Derivative of cos^5x/sin9x

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