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Identical expressions
ln(tg((pi/ four)+(x/ two)))
ln(tg(( Pi divide by 4) plus (x divide by 2)))
ln(tg(( Pi divide by four) plus (x divide by two)))
lntgpi/4+x/2
ln(tg((pi divide by 4)+(x divide by 2)))
Similar expressions
ln(tg((pi/4)-(x/2)))
y=lntg((pi/4)+x/2)
lntg(pi/4+x/2)

The derivative of the function/ ln(tg((pi/4)+(x/2)))

Derivative of ln(tg((pi/4)+(x/2)))

The solution

You have entered [src]

   /   /pi   x\\
log|tan|-- + -||
   \   \4    2//

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

log(tan(pi/4 + x/2))

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2/pi   x\
    tan |-- + -|
1       \4    2/
- + ------------
2        2      
----------------
     /pi   x\   
  tan|-- + -|   
     \4    2/

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

Simplify

The second derivative [src]

                                           2
                       /       2/pi + 2*x\\ 
                       |1 + tan |--------|| 
         2/pi + 2*x\   \        \   4    // 
2 + 2*tan |--------| - ---------------------
          \   4    /          2/pi + 2*x\   
                           tan |--------|   
                               \   4    /   
--------------------------------------------
                     4

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

Simplify

The third derivative [src]

                     /                                      2                         \
                     |                  /       2/pi + 2*x\\      /       2/pi + 2*x\\|
                     |                  |1 + tan |--------||    2*|1 + tan |--------|||
/       2/pi + 2*x\\ |     /pi + 2*x\   \        \   4    //      \        \   4    //|
|1 + tan |--------||*|2*tan|--------| + --------------------- - ----------------------|
\        \   4    // |     \   4    /          3/pi + 2*x\             /pi + 2*x\     |
                     |                      tan |--------|          tan|--------|     |
                     \                          \   4    /             \   4    /     /
---------------------------------------------------------------------------------------
                                           4

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

Simplify

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Derivative of ln(tg((pi/4)+(x/2)))

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