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Derivative of:
Derivative of ln(1+x)
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Derivative of e^(-2*x)
Identical expressions
log(cos(two *x))^(one / two)
logarithm of ( co sinus of e of (2 multiply by x)) to the power of (1 divide by 2)
logarithm of ( co sinus of e of (two multiply by x)) to the power of (one divide by two)
log(cos(2*x))(1/2)
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log(cos(2x))^(1/2)
log(cos(2x))(1/2)
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log(cos(2*x))^(1 divide by 2)

The derivative of the function/ log(cos(2*x))^(1/2)

Derivative of log(cos(2*x))^(1/2)

The solution

You have entered [src]

  _______________
\/ log(cos(2*x))

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

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

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

Transform

Detail solution

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

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

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

The answer is:

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

The graph

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The first derivative [src]

        -sin(2*x)         
--------------------------
           _______________
cos(2*x)*\/ log(cos(2*x))

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

 /                         2                   2                          2           \          
 |          6         8*sin (2*x)         3*sin (2*x)                6*sin (2*x)      |          
-|8 + ------------- + ----------- + ------------------------ + -----------------------|*sin(2*x) 
 |    log(cos(2*x))       2            2         2                2                   |          
 \                     cos (2*x)    cos (2*x)*log (cos(2*x))   cos (2*x)*log(cos(2*x))/          
-------------------------------------------------------------------------------------------------
                                               _______________                                   
                                    cos(2*x)*\/ log(cos(2*x))

$$- \frac{\left(\frac{8 \sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 8 + \frac{6 \sin^{2}{\left(2 x \right)}}{\log{\left(\cos{\left(2 x \right)} \right)} \cos^{2}{\left(2 x \right)}} + \frac{6}{\log{\left(\cos{\left(2 x \right)} \right)}} + \frac{3 \sin^{2}{\left(2 x \right)}}{\log{\left(\cos{\left(2 x \right)} \right)}^{2} \cos^{2}{\left(2 x \right)}}\right) \sin{\left(2 x \right)}}{\sqrt{\log{\left(\cos{\left(2 x \right)} \right)}} \cos{\left(2 x \right)}}$$

Simplify

The graph

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Identical expressions

Derivative of log(cos(2*x))^(1/2)

The graph:

Piecewise:

The solution