Derivative of log2(cos(5x))

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log(cos(5*x))
-------------
    log(2)

$$\frac{\log{\left(\cos{\left(5 x \right)} \right)}}{\log{\left(2 \right)}}$$

d /log(cos(5*x))\
--|-------------|
dx\    log(2)   /

$$\frac{d}{d x} \frac{\log{\left(\cos{\left(5 x \right)} \right)}}{\log{\left(2 \right)}}$$

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \cos{\left(5 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x \right)}$
  The result of the chain rule is:
  
  $- \frac{5 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
So, the result is: $- \frac{5 \sin{\left(5 x \right)}}{\log{\left(2 \right)} \cos{\left(5 x \right)}}$
Now simplify:

$- \frac{5 \tan{\left(5 x \right)}}{\log{\left(2 \right)}}$

The answer is:

$- \frac{5 \tan{\left(5 x \right)}}{\log{\left(2 \right)}}$

The graph

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The first derivative [src]

  -5*sin(5*x)  
---------------
cos(5*x)*log(2)

$$- \frac{5 \sin{\left(5 x \right)}}{\log{\left(2 \right)} \cos{\left(5 x \right)}}$$

Simplify

The second derivative [src]

    /       2     \
    |    sin (5*x)|
-25*|1 + ---------|
    |       2     |
    \    cos (5*x)/
-------------------
       log(2)

$$- \frac{25 \left(\frac{\sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right)}{\log{\left(2 \right)}}$$

Simplify

The third derivative [src]

     /       2     \         
     |    sin (5*x)|         
-250*|1 + ---------|*sin(5*x)
     |       2     |         
     \    cos (5*x)/         
-----------------------------
       cos(5*x)*log(2)

$$- \frac{250 \left(\frac{\sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 1\right) \sin{\left(5 x \right)}}{\log{\left(2 \right)} \cos{\left(5 x \right)}}$$

Simplify

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Identical expressions

Derivative of log2(cos(5x))

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The solution