Derivative of lncos4x

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log(cos(4*x))

$$\log{\left(\cos{\left(4 x \right)} \right)}$$

d                
--(log(cos(4*x)))
dx

$$\frac{d}{d x} \log{\left(\cos{\left(4 x \right)} \right)}$$

Transform

Detail solution

Let $u = \cos{\left(4 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(4 x \right)}$ :
1. Let $u = 4 x$ .
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 $\frac{d}{d x} 4 x$ :
  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: $4$
  The result of the chain rule is:
  
  $- 4 \sin{\left(4 x \right)}$
The result of the chain rule is:

$- \frac{4 \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$
Now simplify:

$- 4 \tan{\left(4 x \right)}$

The answer is:

$- 4 \tan{\left(4 x \right)}$

The graph

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The first derivative [src]

-4*sin(4*x)
-----------
  cos(4*x)

$$- \frac{4 \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$$

Simplify

The second derivative [src]

    /       2     \
    |    sin (4*x)|
-16*|1 + ---------|
    |       2     |
    \    cos (4*x)/

$$- 16 \left(\frac{\sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + 1\right)$$

Simplify

The third derivative [src]

     /       2     \         
     |    sin (4*x)|         
-128*|1 + ---------|*sin(4*x)
     |       2     |         
     \    cos (4*x)/         
-----------------------------
           cos(4*x)

$$- \frac{128 \left(\frac{\sin^{2}{\left(4 x \right)}}{\cos^{2}{\left(4 x \right)}} + 1\right) \sin{\left(4 x \right)}}{\cos{\left(4 x \right)}}$$

Simplify

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Derivative of lncos4x

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The solution