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Identical expressions
log(cos(five *x))^(two)
logarithm of ( co sinus of e of (5 multiply by x)) to the power of (2)
logarithm of ( co sinus of e of (five multiply by x)) to the power of (two)
log(cos(5*x))(2)
logcos5*x2
log(cos(5x))^(2)
log(cos(5x))(2)
logcos5x2
logcos5x^2

The derivative of the function/ log(cos(5*x))^(2)

Derivative of log(cos(5*x))^(2)

The solution

You have entered [src]

   2          
log (cos(5*x))

$$\log{\left(\cos{\left(5 x \right)} \right)}^{2}$$

d /   2          \
--\log (cos(5*x))/
dx

$$\frac{d}{d x} \log{\left(\cos{\left(5 x \right)} \right)}^{2}$$

Transform

Detail solution

Let $u = \log{\left(\cos{\left(5 x \right)} \right)}$ .
Apply the power rule: $u^{2}$ goes to $2 u$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(\cos{\left(5 x \right)} \right)}$ :
1. Let $u = \cos{\left(5 x \right)}$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(5 x \right)}$ :
  1. Let $u = 5 x$ .
  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 $\frac{d}{d x} 5 x$ :
    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: $5$
    The result of the chain rule is:
    
    $- 5 \sin{\left(5 x \right)}$
  The result of the chain rule is:
  
  $- \frac{5 \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
The result of the chain rule is:

$- \frac{10 \log{\left(\cos{\left(5 x \right)} \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$
Now simplify:

$- 10 \log{\left(\cos{\left(5 x \right)} \right)} \tan{\left(5 x \right)}$

The answer is:

$- 10 \log{\left(\cos{\left(5 x \right)} \right)} \tan{\left(5 x \right)}$

The graph

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The first derivative [src]

-10*log(cos(5*x))*sin(5*x)
--------------------------
         cos(5*x)

$$- \frac{10 \log{\left(\cos{\left(5 x \right)} \right)} \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$$

Simplify

The second derivative [src]

   /                    2           2                   \
   |                 sin (5*x)   sin (5*x)*log(cos(5*x))|
50*|-log(cos(5*x)) + --------- - -----------------------|
   |                    2                  2            |
   \                 cos (5*x)          cos (5*x)       /

$$50 \left(- \frac{\log{\left(\cos{\left(5 x \right)} \right)} \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} - \log{\left(\cos{\left(5 x \right)} \right)} + \frac{\sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}}\right)$$

Simplify

The third derivative [src]

    /                           2             2                   \         
    |                      3*sin (5*x)   2*sin (5*x)*log(cos(5*x))|         
250*|3 - 2*log(cos(5*x)) + ----------- - -------------------------|*sin(5*x)
    |                          2                    2             |         
    \                       cos (5*x)            cos (5*x)        /         
----------------------------------------------------------------------------
                                  cos(5*x)

$$\frac{250 \left(- \frac{2 \log{\left(\cos{\left(5 x \right)} \right)} \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} - 2 \log{\left(\cos{\left(5 x \right)} \right)} + \frac{3 \sin^{2}{\left(5 x \right)}}{\cos^{2}{\left(5 x \right)}} + 3\right) \sin{\left(5 x \right)}}{\cos{\left(5 x \right)}}$$

Simplify

The graph

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Derivative of log(cos(5*x))^(2)

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The solution