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Derivative of:
Derivative of sin(x)+cos(x)
Derivative of cos(x)^2
Derivative of sin(x^3)
Derivative of log(x)^2
Identical expressions
- one /(tg(x/ two)+ three)
minus 1 divide by (tg(x divide by 2) plus 3)
minus one divide by (tg(x divide by two) plus three)
-1/tgx/2+3
-1 divide by (tg(x divide by 2)+3)
Similar expressions
1/(tg(x/2)+3)
-1/(tg(x/2)-3)

The derivative of the function/ -1/(tg(x/2)+3)

Derivative of -1/(tg(x/2)+3)

The solution

You have entered [src]

   -1     
----------
   /x\    
tan|-| + 3
   \2/

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

-1/(tan(x/2) + 3)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 /         2/x\\ 
 |      tan |-|| 
 |  1       \2/| 
-|- - - -------| 
 \  2      2   / 
-----------------
              2  
  /   /x\    \   
  |tan|-| + 3|   
  \   \2/    /

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

              /                               2                         \
              |                  /       2/x\\      /       2/x\\    /x\|
              |                3*|1 + tan |-||    6*|1 + tan |-||*tan|-||
/       2/x\\ |         2/x\     \        \2//      \        \2//    \2/|
|1 + tan |-||*|1 + 3*tan |-| + ---------------- - ----------------------|
\        \2// |          \2/                2                  /x\      |
              |                 /       /x\\            3 + tan|-|      |
              |                 |3 + tan|-||                   \2/      |
              \                 \       \2//                            /
-------------------------------------------------------------------------
                                           2                             
                               /       /x\\                              
                             4*|3 + tan|-||                              
                               \       \2//

$$\frac{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \left(3 \tan^{2}{\left(\frac{x}{2} \right)} + 1 - \frac{6 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right) \tan{\left(\frac{x}{2} \right)}}{\tan{\left(\frac{x}{2} \right)} + 3} + \frac{3 \left(\tan^{2}{\left(\frac{x}{2} \right)} + 1\right)^{2}}{\left(\tan{\left(\frac{x}{2} \right)} + 3\right)^{2}}\right)}{4 \left(\tan{\left(\frac{x}{2} \right)} + 3\right)^{2}}$$

Simplify

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Derivative of -1/(tg(x/2)+3)

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The solution