Derivative of y=lncos²5x

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   25        
log  (cos(x))

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

log(cos(x))^25

Detail solution

Let $u = \log{\left(\cos{\left(x \right)} \right)}$ .
Apply the power rule: $u^{25}$ goes to $25 u^{24}$
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:

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

$- 25 \log{\left(\cos{\left(x \right)} \right)}^{24} \tan{\left(x \right)}$

The answer is:

$- 25 \log{\left(\cos{\left(x \right)} \right)}^{24} \tan{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                 /                     2         2               \
      23         |               24*sin (x)   sin (x)*log(cos(x))|
25*log  (cos(x))*|-log(cos(x)) + ---------- - -------------------|
                 |                   2                 2         |
                 \                cos (x)           cos (x)      /

$$25 \left(- \frac{\log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \log{\left(\cos{\left(x \right)} \right)} + \frac{24 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{23}$$

Simplify

The third derivative [src]

                 /                                         2         2            2            2               \       
      22         |     2                            276*sin (x)   log (cos(x))*sin (x)   36*sin (x)*log(cos(x))|       
50*log  (cos(x))*|- log (cos(x)) + 36*log(cos(x)) - ----------- - -------------------- + ----------------------|*sin(x)
                 |                                       2                 2                       2           |       
                 \                                    cos (x)           cos (x)                 cos (x)        /       
-----------------------------------------------------------------------------------------------------------------------
                                                         cos(x)

$$\frac{50 \left(- \frac{\log{\left(\cos{\left(x \right)} \right)}^{2} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \log{\left(\cos{\left(x \right)} \right)}^{2} + \frac{36 \log{\left(\cos{\left(x \right)} \right)} \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 36 \log{\left(\cos{\left(x \right)} \right)} - \frac{276 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}\right) \log{\left(\cos{\left(x \right)} \right)}^{22} \sin{\left(x \right)}}{\cos{\left(x \right)}}$$

Simplify

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Derivative of y=lncos²5x

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