Derivative of y=ln(sin12x)

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log(sin(12*x))

$$\log{\left(\sin{\left(12 x \right)} \right)}$$

log(sin(12*x))

Detail solution

Let $u = \sin{\left(12 x \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(12 x \right)}$ :
1. Let $u = 12 x$ .
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 $\frac{d}{d x} 12 x$ :
  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: $12$
  The result of the chain rule is:
  
  $12 \cos{\left(12 x \right)}$
The result of the chain rule is:

$\frac{12 \cos{\left(12 x \right)}}{\sin{\left(12 x \right)}}$
Now simplify:

$\frac{12}{\tan{\left(12 x \right)}}$

The answer is:

$\frac{12}{\tan{\left(12 x \right)}}$

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The first derivative [src]

12*cos(12*x)
------------
 sin(12*x)

$$\frac{12 \cos{\left(12 x \right)}}{\sin{\left(12 x \right)}}$$

Simplify

The second derivative [src]

     /       2      \
     |    cos (12*x)|
-144*|1 + ----------|
     |       2      |
     \    sin (12*x)/

$$- 144 \left(1 + \frac{\cos^{2}{\left(12 x \right)}}{\sin^{2}{\left(12 x \right)}}\right)$$

Simplify

The third derivative [src]

     /       2      \          
     |    cos (12*x)|          
3456*|1 + ----------|*cos(12*x)
     |       2      |          
     \    sin (12*x)/          
-------------------------------
           sin(12*x)

$$\frac{3456 \left(1 + \frac{\cos^{2}{\left(12 x \right)}}{\sin^{2}{\left(12 x \right)}}\right) \cos{\left(12 x \right)}}{\sin{\left(12 x \right)}}$$

Simplify

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Derivative of y=ln(sin12x)

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