Derivative of ln(sin(4*x))

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log(sin(4*x))

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

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(sin(4*x))

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The solution