Derivative of lncos(3x^2)

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   /   /   2\\
log\cos\3*x //

$$\log{\left(\cos{\left(3 x^{2} \right)} \right)}$$

log(cos(3*x^2))

Detail solution

Let $u = \cos{\left(3 x^{2} \right)}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(3 x^{2} \right)}$ :
1. Let $u = 3 x^{2}$ .
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 $\frac{d}{d x} 3 x^{2}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $6 x$
  The result of the chain rule is:
  
  $- 6 x \sin{\left(3 x^{2} \right)}$
The result of the chain rule is:

$- \frac{6 x \sin{\left(3 x^{2} \right)}}{\cos{\left(3 x^{2} \right)}}$
Now simplify:

$- 6 x \tan{\left(3 x^{2} \right)}$

The answer is:

$- 6 x \tan{\left(3 x^{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        /   2\
-6*x*sin\3*x /
--------------
     /   2\   
  cos\3*x /

$$- \frac{6 x \sin{\left(3 x^{2} \right)}}{\cos{\left(3 x^{2} \right)}}$$

Simplify

The second derivative [src]

   /          /   2\      2    2/   2\\
   |   2   sin\3*x /   6*x *sin \3*x /|
-6*|6*x  + --------- + ---------------|
   |          /   2\         2/   2\  |
   \       cos\3*x /      cos \3*x /  /

$$- 6 \left(\frac{6 x^{2} \sin^{2}{\left(3 x^{2} \right)}}{\cos^{2}{\left(3 x^{2} \right)}} + 6 x^{2} + \frac{\sin{\left(3 x^{2} \right)}}{\cos{\left(3 x^{2} \right)}}\right)$$

Simplify

The third derivative [src]

       /       2/   2\      2    /   2\      2    3/   2\\
       |    sin \3*x /   4*x *sin\3*x /   4*x *sin \3*x /|
-108*x*|1 + ---------- + -------------- + ---------------|
       |       2/   2\        /   2\            3/   2\  |
       \    cos \3*x /     cos\3*x /         cos \3*x /  /

$$- 108 x \left(\frac{4 x^{2} \sin^{3}{\left(3 x^{2} \right)}}{\cos^{3}{\left(3 x^{2} \right)}} + \frac{4 x^{2} \sin{\left(3 x^{2} \right)}}{\cos{\left(3 x^{2} \right)}} + \frac{\sin^{2}{\left(3 x^{2} \right)}}{\cos^{2}{\left(3 x^{2} \right)}} + 1\right)$$

Simplify

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Identical expressions

Derivative of lncos(3x^2)

The graph:

Piecewise:

The solution