Derivative of ln(sin(3x))

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log(sin(3*x))

$$\log{\left(\sin{\left(3 x \right)} \right)}$$

log(sin(3*x))

Detail solution

Let $u = \sin{\left(3 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(3 x \right)}$ :
1. Let $u = 3 x$ .
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 $\frac{d}{d x} 3 x$ :
  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: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x \right)}$
The result of the chain rule is:

$\frac{3 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$
Now simplify:

$\frac{3}{\tan{\left(3 x \right)}}$

The answer is:

$\frac{3}{\tan{\left(3 x \right)}}$

The graph

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The first derivative [src]

3*cos(3*x)
----------
 sin(3*x)

$$\frac{3 \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$$

Simplify

The second derivative [src]

   /       2     \
   |    cos (3*x)|
-9*|1 + ---------|
   |       2     |
   \    sin (3*x)/

$$- 9 \left(1 + \frac{\cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right)$$

Simplify

The third derivative [src]

   /       2     \         
   |    cos (3*x)|         
54*|1 + ---------|*cos(3*x)
   |       2     |         
   \    sin (3*x)/         
---------------------------
          sin(3*x)

$$\frac{54 \left(1 + \frac{\cos^{2}{\left(3 x \right)}}{\sin^{2}{\left(3 x \right)}}\right) \cos{\left(3 x \right)}}{\sin{\left(3 x \right)}}$$

Simplify

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Derivative of ln(sin(3x))

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The solution