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Similar expressions
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The derivative of the function/ 3lnsqrt(cos(2*x))

Derivative of 3lnsqrt(cos(2*x))

The solution

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     /  __________\
3*log\\/ cos(2*x) /

$$3 \log{\left(\sqrt{\cos{\left(2 x \right)}} \right)}$$

3*log(sqrt(cos(2*x)))

Detail solution

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

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

The answer is:

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

The graph

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The first derivative [src]

-3*sin(2*x)
-----------
  cos(2*x)

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

Simplify

The second derivative [src]

   /       2     \
   |    sin (2*x)|
-6*|1 + ---------|
   |       2     |
   \    cos (2*x)/

$$- 6 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right)$$

Simplify

The third derivative [src]

    /       2     \         
    |    sin (2*x)|         
-24*|1 + ---------|*sin(2*x)
    |       2     |         
    \    cos (2*x)/         
----------------------------
          cos(2*x)

$$- \frac{24 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$$

Simplify

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Derivative of 3lnsqrt(cos(2*x))

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The solution