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Identical expressions
log(cos(x^ three))
logarithm of ( co sinus of e of (x cubed ))
logarithm of ( co sinus of e of (x to the power of three))
log(cos(x3))
logcosx3
log(cos(x³))
log(cos(x to the power of 3))
logcosx^3

The derivative of the function/ log(cos(x^3))

Derivative of log(cos(x^3))

The solution

You have entered [src]

   /   / 3\\
log\cos\x //

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

d /   /   / 3\\\
--\log\cos\x ///
dx

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

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 /  /

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

Simplify

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 /  /

$$- 6 \cdot \left(\frac{9 x^{6} \sin^{3}{\left(x^{3} \right)}}{\cos^{3}{\left(x^{3} \right)}} + \frac{9 x^{6} \sin{\left(x^{3} \right)}}{\cos{\left(x^{3} \right)}} + \frac{9 x^{3} \sin^{2}{\left(x^{3} \right)}}{\cos^{2}{\left(x^{3} \right)}} + 9 x^{3} + \frac{\sin{\left(x^{3} \right)}}{\cos{\left(x^{3} \right)}}\right)$$

Simplify

The graph

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Identical expressions

Derivative of log(cos(x^3))

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The solution