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Identical expressions
y=ln(tg2x+ one / four)
y equally ln(tg2x plus 1 divide by 4)
y equally ln(tg2x plus one divide by four)
y=lntg2x+1/4
y=ln(tg2x+1 divide by 4)
Similar expressions
y=ln(tg2x-1/4)
ln*tg(2x+1/4)

The derivative of the function/ y=ln(tg2x+1/4)

Derivative of y=ln(tg2x+1/4)

The solution

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log(tan(2*x) + 1/4)

$$\log{\left(\tan{\left(2 x \right)} + \frac{1}{4} \right)}$$

log(tan(2*x) + 1/4)

Detail solution

Let $u = \tan{\left(2 x \right)} + \frac{1}{4}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\tan{\left(2 x \right)} + \frac{1}{4}\right)$ :
1. Differentiate $\tan{\left(2 x \right)} + \frac{1}{4}$ term by term:
  1. Rewrite the function to be differentiated:
    
    $\tan{\left(2 x \right)} = \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \cos{\left(2 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 2 x$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    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:
      
      $2 \cos{\left(2 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    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:
      
      $- 2 \sin{\left(2 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
  3. The derivative of the constant $\frac{1}{4}$ is zero.
  The result is: $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
The result of the chain rule is:

$\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\left(\tan{\left(2 x \right)} + \frac{1}{4}\right) \cos^{2}{\left(2 x \right)}}$
Now simplify:

$\frac{16}{4 \sin{\left(4 x \right)} + \cos{\left(4 x \right)} + 1}$

The answer is:

$\frac{16}{4 \sin{\left(4 x \right)} + \cos{\left(4 x \right)} + 1}$

The first derivative [src]

         2     
2 + 2*tan (2*x)
---------------
 tan(2*x) + 1/4

$$\frac{2 \tan^{2}{\left(2 x \right)} + 2}{\tan{\left(2 x \right)} + \frac{1}{4}}$$

Simplify

The second derivative [src]

                   /    /       2     \           \
   /       2     \ |  2*\1 + tan (2*x)/           |
32*\1 + tan (2*x)/*|- ----------------- + tan(2*x)|
                   \    1 + 4*tan(2*x)            /
---------------------------------------------------
                   1 + 4*tan(2*x)

$$\frac{32 \left(\tan{\left(2 x \right)} - \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{4 \tan{\left(2 x \right)} + 1}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{4 \tan{\left(2 x \right)} + 1}$$

Simplify

The third derivative [src]

                   /                                    2                              \
                   |                     /       2     \       /       2     \         |
   /       2     \ |         2        16*\1 + tan (2*x)/    12*\1 + tan (2*x)/*tan(2*x)|
64*\1 + tan (2*x)/*|1 + 3*tan (2*x) + ------------------- - ---------------------------|
                   |                                   2           1 + 4*tan(2*x)      |
                   \                   (1 + 4*tan(2*x))                                /
----------------------------------------------------------------------------------------
                                     1 + 4*tan(2*x)

$$\frac{64 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(3 \tan^{2}{\left(2 x \right)} + 1 - \frac{12 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(2 x \right)}}{4 \tan{\left(2 x \right)} + 1} + \frac{16 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}}{\left(4 \tan{\left(2 x \right)} + 1\right)^{2}}\right)}{4 \tan{\left(2 x \right)} + 1}$$

Simplify

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Derivative of y=ln(tg2x+1/4)

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