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Derivative of:
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Derivative of cos^6(x)
Derivative of y=lnctg((x/3)+(pi/4))
Derivative of log7(x)
Identical expressions
y=lnctg((x/ three)+(pi/ four))
y equally lnctg((x divide by 3) plus ( Pi divide by 4))
y equally lnctg((x divide by three) plus ( Pi divide by four))
y=lnctgx/3+pi/4
y=lnctg((x divide by 3)+(pi divide by 4))
Similar expressions
y=lnctg((x/3)-(pi/4))

The derivative of the function/ y=lnctg((x/3)+(pi/4))

Derivative of y=lnctg((x/3)+(pi/4))

The solution

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   /   /x   pi\\
log|cot|- + --||
   \   \3   4 //

$$\log{\left(\cot{\left(\frac{x}{3} + \frac{\pi}{4} \right)} \right)}$$

log(cot(x/3 + pi/4))

Detail solution

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

$- \frac{\frac{\sin^{2}{\left(\frac{x}{3} + \frac{\pi}{4} \right)}}{3} + \frac{\cos^{2}{\left(\frac{x}{3} + \frac{\pi}{4} \right)}}{3}}{\cos^{2}{\left(\frac{x}{3} + \frac{\pi}{4} \right)} \tan^{2}{\left(\frac{x}{3} + \frac{\pi}{4} \right)} \cot{\left(\frac{x}{3} + \frac{\pi}{4} \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

         2/x   pi\
      cot |- + --|
  1       \3   4 /
- - - ------------
  3        3      
------------------
      /x   pi\    
   cot|- + --|    
      \3   4 /

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

Simplify

The second derivative [src]

                                               2
                         /       2/3*pi + 4*x\\ 
                         |1 + cot |----------|| 
         2/3*pi + 4*x\   \        \    12    // 
2 + 2*cot |----------| - -----------------------
          \    12    /          2/3*pi + 4*x\   
                             cot |----------|   
                                 \    12    /   
------------------------------------------------
                       9

$$\frac{- \frac{\left(\cot^{2}{\left(\frac{4 x + 3 \pi}{12} \right)} + 1\right)^{2}}{\cot^{2}{\left(\frac{4 x + 3 \pi}{12} \right)}} + 2 \cot^{2}{\left(\frac{4 x + 3 \pi}{12} \right)} + 2}{9}$$

Simplify

The third derivative [src]

                         /                                            2                           \
                         |                      /       2/3*pi + 4*x\\      /       2/3*pi + 4*x\\|
                         |                      |1 + cot |----------||    2*|1 + cot |----------|||
  /       2/3*pi + 4*x\\ |       /3*pi + 4*x\   \        \    12    //      \        \    12    //|
2*|1 + cot |----------||*|- 2*cot|----------| - ----------------------- + ------------------------|
  \        \    12    // |       \    12    /          3/3*pi + 4*x\             /3*pi + 4*x\     |
                         |                          cot |----------|          cot|----------|     |
                         \                              \    12    /             \    12    /     /
---------------------------------------------------------------------------------------------------
                                                 27

$$\frac{2 \left(\cot^{2}{\left(\frac{4 x + 3 \pi}{12} \right)} + 1\right) \left(- \frac{\left(\cot^{2}{\left(\frac{4 x + 3 \pi}{12} \right)} + 1\right)^{2}}{\cot^{3}{\left(\frac{4 x + 3 \pi}{12} \right)}} + \frac{2 \left(\cot^{2}{\left(\frac{4 x + 3 \pi}{12} \right)} + 1\right)}{\cot{\left(\frac{4 x + 3 \pi}{12} \right)}} - 2 \cot{\left(\frac{4 x + 3 \pi}{12} \right)}\right)}{27}$$

Simplify

The graph

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Identical expressions

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Derivative of y=lnctg((x/3)+(pi/4))

The graph:

Piecewise:

The solution

Method #1

Method #2