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log(cos(x^3))

Derivative of log(cos(x^3))

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
   /   / 3\\
log\cos\x //
$$\log{\left(\cos{\left(x^{3} \right)} \right)}$$
d /   /   / 3\\\
--\log\cos\x ///
dx              
$$\frac{d}{d x} \log{\left(\cos{\left(x^{3} \right)} \right)}$$
Detail solution
  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Let .

    2. The derivative of cosine is negative sine:

    3. Then, apply the chain rule. Multiply by :

      1. Apply the power rule: goes to

      The result of the chain rule is:

    The result of the chain rule is:

  4. Now simplify:


The answer is:

The graph
The first derivative [src]
    2    / 3\
-3*x *sin\x /
-------------
      / 3\   
   cos\x /   
$$- \frac{3 x^{2} \sin{\left(x^{3} \right)}}{\cos{\left(x^{3} \right)}}$$
The second derivative [src]
     /            / 3\      3    2/ 3\\
     |   3   2*sin\x /   3*x *sin \x /|
-3*x*|3*x  + --------- + -------------|
     |           / 3\          2/ 3\  |
     \        cos\x /       cos \x /  /
$$- 3 x \left(\frac{3 x^{3} \sin^{2}{\left(x^{3} \right)}}{\cos^{2}{\left(x^{3} \right)}} + 3 x^{3} + \frac{2 \sin{\left(x^{3} \right)}}{\cos{\left(x^{3} \right)}}\right)$$
The third derivative [src]
   /          / 3\      3    2/ 3\      6    / 3\      6    3/ 3\\
   |   3   sin\x /   9*x *sin \x /   9*x *sin\x /   9*x *sin \x /|
-6*|9*x  + ------- + ------------- + ------------ + -------------|
   |          / 3\         2/ 3\          / 3\            3/ 3\  |
   \       cos\x /      cos \x /       cos\x /         cos \x /  /
$$- 6 \cdot \left(\frac{9 x^{6} \sin^{3}{\left(x^{3} \right)}}{\cos^{3}{\left(x^{3} \right)}} + \frac{9 x^{6} \sin{\left(x^{3} \right)}}{\cos{\left(x^{3} \right)}} + \frac{9 x^{3} \sin^{2}{\left(x^{3} \right)}}{\cos^{2}{\left(x^{3} \right)}} + 9 x^{3} + \frac{\sin{\left(x^{3} \right)}}{\cos{\left(x^{3} \right)}}\right)$$
The graph
Derivative of log(cos(x^3))