Derivative of cos²5x/sin²5x

The solution

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   25   
cos  (x)
--------
   25   
sin  (x)

$$\frac{\cos^{25}{\left(x \right)}}{\sin^{25}{\left(x \right)}}$$

cos(x)^25/sin(x)^25

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos^{25}{\left(x \right)}$ and $g{\left(x \right)} = \sin^{25}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \cos{\left(x \right)}$ .
2. Apply the power rule: $u^{25}$ goes to $25 u^{24}$
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:
  
  $- 25 \sin{\left(x \right)} \cos^{24}{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{25}$ goes to $25 u^{24}$
3. 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:
  
  $25 \sin^{24}{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{- 25 \sin^{26}{\left(x \right)} \cos^{24}{\left(x \right)} - 25 \sin^{24}{\left(x \right)} \cos^{26}{\left(x \right)}}{\sin^{50}{\left(x \right)}}$
Now simplify:

$- \frac{25 \cos^{24}{\left(x \right)}}{\sin^{26}{\left(x \right)}}$

The answer is:

$- \frac{25 \cos^{24}{\left(x \right)}}{\sin^{26}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        26            24          
  25*cos  (x)   25*cos  (x)*sin(x)
- ----------- - ------------------
       26               25        
    sin  (x)         sin  (x)

$$- \frac{25 \sin{\left(x \right)} \cos^{24}{\left(x \right)}}{\sin^{25}{\left(x \right)}} - \frac{25 \cos^{26}{\left(x \right)}}{\sin^{26}{\left(x \right)}}$$

Simplify

The second derivative [src]

            /                                  /          2   \\
      23    |      2            2         2    |    26*cos (x)||
25*cos  (x)*|24*sin (x) + 49*cos (x) + cos (x)*|1 + ----------||
            |                                  |        2     ||
            \                                  \     sin (x)  //
----------------------------------------------------------------
                               25                               
                            sin  (x)

$$\frac{25 \left(\left(1 + \frac{26 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)} + 24 \sin^{2}{\left(x \right)} + 49 \cos^{2}{\left(x \right)}\right) \cos^{23}{\left(x \right)}}{\sin^{25}{\left(x \right)}}$$

Simplify

The third derivative [src]

             /                                                                   /            2   \                                      \
             |                                                              4    |     702*cos (x)|                                      |
             |                                                           cos (x)*|77 + -----------|                                      |
             |                                        /          2   \           |          2     |         2    /     2            2   \|
       22    |        2             2            2    |    26*cos (x)|           \       sin (x)  /   75*cos (x)*\- cos (x) + 24*sin (x)/|
-25*cos  (x)*|- 73*cos (x) + 552*sin (x) + 75*cos (x)*|1 + ----------| + -------------------------- + -----------------------------------|
             |                                        |        2     |               2                                 2                 |
             \                                        \     sin (x)  /            sin (x)                           sin (x)              /
------------------------------------------------------------------------------------------------------------------------------------------
                                                                    24                                                                    
                                                                 sin  (x)

$$- \frac{25 \left(75 \left(1 + \frac{26 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos^{2}{\left(x \right)} + \frac{\left(77 + \frac{702 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \cos^{4}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{75 \left(24 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 552 \sin^{2}{\left(x \right)} - 73 \cos^{2}{\left(x \right)}\right) \cos^{22}{\left(x \right)}}{\sin^{24}{\left(x \right)}}$$

Simplify

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Derivative of cos²5x/sin²5x

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The solution