Derivative of y=lncos²5x/sin³5x

The solution

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   25        
log  (cos(x))
-------------
      35     
   sin  (x)

$$\frac{\log{\left(\cos{\left(x \right)} \right)}^{25}}{\sin^{35}{\left(x \right)}}$$

log(cos(x))^25/sin(x)^35

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(\cos{\left(x \right)} \right)}^{25}$ and $g{\left(x \right)} = \sin^{35}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = \log{\left(\cos{\left(x \right)} \right)}$ .
2. Apply the power rule: $u^{25}$ goes to $25 u^{24}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\cos{\left(x \right)} \right)}$ :
  1. Let $u = \cos{\left(x \right)}$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  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:
    
    $- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  The result of the chain rule is:
  
  $- \frac{25 \log{\left(\cos{\left(x \right)} \right)}^{24} \sin{\left(x \right)}}{\cos{\left(x \right)}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{35}$ goes to $35 u^{34}$
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:
  
  $35 \sin^{34}{\left(x \right)} \cos{\left(x \right)}$
Now plug in to the quotient rule:

$\frac{- 35 \log{\left(\cos{\left(x \right)} \right)}^{25} \sin^{34}{\left(x \right)} \cos{\left(x \right)} - \frac{25 \log{\left(\cos{\left(x \right)} \right)}^{24} \sin^{36}{\left(x \right)}}{\cos{\left(x \right)}}}{\sin^{70}{\left(x \right)}}$
Now simplify:

$\frac{\left(- 35 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + 25 \cos{\left(x \right)} - \frac{25}{\cos{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{24}}{\sin^{36}{\left(x \right)}}$

The answer is:

$\frac{\left(- 35 \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} + 25 \cos{\left(x \right)} - \frac{25}{\cos{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{24}}{\sin^{36}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        25                        24               
  35*log  (cos(x))*cos(x)   25*log  (cos(x))*sin(x)
- ----------------------- - -----------------------
             36                           35       
          sin  (x)              cos(x)*sin  (x)

$$- \frac{35 \log{\left(\cos{\left(x \right)} \right)}^{25} \cos{\left(x \right)}}{\sin^{36}{\left(x \right)}} - \frac{25 \log{\left(\cos{\left(x \right)} \right)}^{24} \sin{\left(x \right)}}{\sin^{35}{\left(x \right)} \cos{\left(x \right)}}$$

Simplify

The second derivative [src]

                /                                 /          2   \          2           2               \
     23         |                       2         |    36*cos (x)|   120*sin (x)   5*sin (x)*log(cos(x))|
5*log  (cos(x))*|345*log(cos(x)) + 7*log (cos(x))*|1 + ----------| + ----------- - ---------------------|
                |                                 |        2     |        2                  2          |
                \                                 \     sin (x)  /     cos (x)            cos (x)       /
---------------------------------------------------------------------------------------------------------
                                                    35                                                   
                                                 sin  (x)

$$\frac{5 \left(7 \left(1 + \frac{36 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{2} - \frac{5 \log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 345 \log{\left(\cos{\left(x \right)} \right)} + \frac{120 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{23}}{\sin^{35}{\left(x \right)}}$$

Simplify

The third derivative [src]

                /     /                                       2         2            2            2               \                    /          2   \                  /              2   \              /        2         2                             \                   \
                |     |   2                            276*sin (x)   log (cos(x))*sin (x)   36*sin (x)*log(cos(x))|          2         |    36*cos (x)|        3         |      1332*cos (x)|              |  24*sin (x)   sin (x)*log(cos(x))              |                   |
                |  10*|log (cos(x)) - 36*log(cos(x)) + ----------- + -------------------- - ----------------------|   525*log (cos(x))*|1 + ----------|   7*log (cos(x))*|107 + ------------|*cos(x)   525*|- ---------- + ------------------- + log(cos(x))|*cos(x)*log(cos(x))|
                |     |                                     2                 2                       2           |                    |        2     |                  |           2      |              |      2                 2                       |                   |
     22         |     \                                  cos (x)           cos (x)                 cos (x)        /                    \     sin (x)  /                  \        sin (x)   /              \   cos (x)           cos (x)                    /                   |
5*log  (cos(x))*|- ------------------------------------------------------------------------------------------------ - --------------------------------- - ------------------------------------------ + -------------------------------------------------------------------------|
                |                                               cos(x)                                                              cos(x)                                    2                                                            2                                    |
                \                                                                                                                                                          sin (x)                                                      sin (x)                                 /
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                        34                                                                                                                                       
                                                                                                                                     sin  (x)

$$\frac{5 \left(- \frac{525 \left(1 + \frac{36 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{2}}{\cos{\left(x \right)}} - \frac{7 \left(107 + \frac{1332 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{3} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{525 \left(\frac{\log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)} - \frac{24 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{10 \left(\frac{\log{\left(\cos{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \log{\left(\cos{\left(x \right)} \right)}^{2} - \frac{36 \log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - 36 \log{\left(\cos{\left(x \right)} \right)} + \frac{276 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right)}{\cos{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{22}}{\sin^{34}{\left(x \right)}}$$

Simplify

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Derivative of y=lncos²5x/sin³5x

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