Derivative of y=ln*cos²5x

The solution

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          25   
log(x)*cos  (x)

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

log(x)*cos(x)^25

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
$g{\left(x \right)} = \cos^{25}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
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:
  
  $- 25 \sin{\left(x \right)} \cos^{24}{\left(x \right)}$
The result is: $- 25 \log{\left(x \right)} \sin{\left(x \right)} \cos^{24}{\left(x \right)} + \frac{\cos^{25}{\left(x \right)}}{x}$
Now simplify:

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

         /     2                                                           \
   23    |  cos (x)      /     2            2   \          50*cos(x)*sin(x)|
cos  (x)*|- ------- + 25*\- cos (x) + 24*sin (x)/*log(x) - ----------------|
         |      2                                                 x        |
         \     x                                                           /

$$\left(25 \left(24 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} - \frac{50 \sin{\left(x \right)} \cos{\left(x \right)}}{x} - \frac{\cos^{2}{\left(x \right)}}{x^{2}}\right) \cos^{23}{\left(x \right)}$$

Simplify

The third derivative [src]

         /     3                                                         /     2            2   \                2          \
   22    |2*cos (x)      /        2             2   \                 75*\- cos (x) + 24*sin (x)/*cos(x)   75*cos (x)*sin(x)|
cos  (x)*|--------- - 25*\- 73*cos (x) + 552*sin (x)/*log(x)*sin(x) + ---------------------------------- + -----------------|
         |     3                                                                      x                             2       |
         \    x                                                                                                    x        /

$$\left(- 25 \left(552 \sin^{2}{\left(x \right)} - 73 \cos^{2}{\left(x \right)}\right) \log{\left(x \right)} \sin{\left(x \right)} + \frac{75 \left(24 \sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)}}{x} + \frac{75 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{x^{2}} + \frac{2 \cos^{3}{\left(x \right)}}{x^{3}}\right) \cos^{22}{\left(x \right)}$$

Simplify

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Derivative of y=ln*cos²5x

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