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Derivative of:
Derivative of x^4*e^x
Derivative of x^2*sin(2*x-3)
Derivative of x^2/cosx
Derivative of sin(x)+cot(x)
Identical expressions
ln^ two (cos*x)
ln squared ( co sinus of e of multiply by x)
ln to the power of two ( co sinus of e of multiply by x)
ln2(cos*x)
ln2cos*x
ln²(cos*x)
ln to the power of 2(cos*x)
ln^2(cosx)
ln2(cosx)
ln2cosx
ln^2cosx

The derivative of the function/ ln^2(cos*x)

Derivative of ln^2(cos*x)

The solution

You have entered [src]

   2        
log (cos(x))

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

log(cos(x))^2

Detail solution

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

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

$- 2 \log{\left(\cos{\left(x \right)} \right)} \tan{\left(x \right)}$

The answer is:

$- 2 \log{\left(\cos{\left(x \right)} \right)} \tan{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-2*log(cos(x))*sin(x)
---------------------
        cos(x)

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

Simplify

The second derivative [src]

  /                  2         2               \
  |               sin (x)   sin (x)*log(cos(x))|
2*|-log(cos(x)) + ------- - -------------------|
  |                  2               2         |
  \               cos (x)         cos (x)      /

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of ln^2(cos*x)

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The solution