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Derivative of:
Derivative of (x+5)/(x+1)
Derivative of -x/2
Derivative of sin(4x)
Derivative of x*2^x
Identical expressions
(cos2x+sqrtsinx- one)/(tgx-sqrt3)
( co sinus of e of 2x plus square root of sinus of x minus 1) divide by (tgx minus square root of 3)
( co sinus of e of 2x plus square root of sinus of x minus one) divide by (tgx minus square root of 3)
(cos2x+√sinx-1)/(tgx-√3)
cos2x+sqrtsinx-1/tgx-sqrt3
(cos2x+sqrtsinx-1) divide by (tgx-sqrt3)
Similar expressions
(cos2x+sqrtsinx+1)/(tgx-sqrt3)
(cos2x+sqrtsinx-1)/(tgx+sqrt3)
(cos2x-sqrtsinx-1)/(tgx-sqrt3)

The derivative of the function/ (cos2x+sqrtsinx-1)/(tgx-sqrt3)

Derivative of (cos2x+sqrtsinx-1)/(tgx-sqrt3)

The solution

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             ________    
cos(2*x) + \/ sin(x)  - 1
-------------------------
                 ___     
      tan(x) - \/ 3

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

(cos(2*x) + sqrt(sin(x)) - 1)/(tan(x) - sqrt(3))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \sqrt{\sin{\left(x \right)}} + \cos{\left(2 x \right)} - 1$ and $g{\left(x \right)} = \tan{\left(x \right)} - \sqrt{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\sqrt{\sin{\left(x \right)}} + \cos{\left(2 x \right)} - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Let $u = \sin{\left(x \right)}$ .
  3. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $\frac{\cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}$
  5. Let $u = 2 x$ .
  6. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  7. 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.
      1. 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 is: $- 2 \sin{\left(2 x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $\tan{\left(x \right)} - \sqrt{3}$ term by term:
  1. The derivative of the constant $- \sqrt{3}$ is zero.
  2. Rewrite the function to be differentiated:
    
    $\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}$
  3. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = \cos{\left(x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
  The result is: $\frac{\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}}$
Now plug in to the quotient rule:

$\frac{- \frac{\left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right) \left(\sqrt{\sin{\left(x \right)}} + \cos{\left(2 x \right)} - 1\right)}{\cos^{2}{\left(x \right)}} + \left(- 2 \sin{\left(2 x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{\sin{\left(x \right)}}}\right) \left(\tan{\left(x \right)} - \sqrt{3}\right)}{\left(\tan{\left(x \right)} - \sqrt{3}\right)^{2}}$
Now simplify:

$\frac{- \frac{\left(8 \sin^{\frac{3}{2}}{\left(x \right)} - 1\right) \left(\tan{\left(x \right)} - \sqrt{3}\right) \cos^{3}{\left(x \right)}}{2} + 2 \sin^{\frac{5}{2}}{\left(x \right)} - \sin{\left(x \right)}}{\left(\tan{\left(x \right)} - \sqrt{3}\right)^{2} \sqrt{\sin{\left(x \right)}} \cos^{2}{\left(x \right)}}$

The answer is:

$\frac{- \frac{\left(8 \sin^{\frac{3}{2}}{\left(x \right)} - 1\right) \left(\tan{\left(x \right)} - \sqrt{3}\right) \cos^{3}{\left(x \right)}}{2} + 2 \sin^{\frac{5}{2}}{\left(x \right)} - \sin{\left(x \right)}}{\left(\tan{\left(x \right)} - \sqrt{3}\right)^{2} \sqrt{\sin{\left(x \right)}} \cos^{2}{\left(x \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                 cos(x)                                                
-2*sin(2*x) + ------------                                             
                  ________   /        2   \ /             ________    \
              2*\/ sin(x)    \-1 - tan (x)/*\cos(2*x) + \/ sin(x)  - 1/
-------------------------- + ------------------------------------------
                 ___                                     2             
      tan(x) - \/ 3                      /           ___\              
                                         \tan(x) - \/ 3 /

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

Simplify

The second derivative [src]

                                                                                                   /           2               \                             
                                         /       2   \ /               cos(x)  \     /       2   \ |    1 + tan (x)            | /       ________           \
                                         \1 + tan (x)/*|4*sin(2*x) - ----------|   2*\1 + tan (x)/*|- ---------------- + tan(x)|*\-1 + \/ sin(x)  + cos(2*x)/
                ________        2                      |               ________|                   |      ___                  |                             
              \/ sin(x)      cos (x)                   \             \/ sin(x) /                   \  - \/ 3  + tan(x)         /                             
-4*cos(2*x) - ---------- - ----------- + --------------------------------------- - --------------------------------------------------------------------------
                  2             3/2                      ___                                                        ___                                      
                           4*sin   (x)               - \/ 3  + tan(x)                                           - \/ 3  + tan(x)                             
-------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                           ___                                                                               
                                                                       - \/ 3  + tan(x)

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

Simplify

The third derivative [src]

                                                                                                                                                  /                                 2                          \                                                                          
                                                                                                                                                  |                    /       2   \       /       2   \       |                                                                          
                                                          /                                 2    \     /       2   \ /       ________           \ |         2        3*\1 + tan (x)/     6*\1 + tan (x)/*tan(x)|                                             /           2               \
                                            /       2   \ |    ________                  cos (x) |   2*\1 + tan (x)/*\-1 + \/ sin(x)  + cos(2*x)/*|1 + 3*tan (x) + ------------------- - ----------------------|     /       2   \ /               cos(x)  \ |    1 + tan (x)            |
                                          3*\1 + tan (x)/*|2*\/ sin(x)  + 16*cos(2*x) + ---------|                                                |                                  2          ___            |   3*\1 + tan (x)/*|4*sin(2*x) - ----------|*|- ---------------- + tan(x)|
                                  3                       |                                3/2   |                                                |                /    ___         \       - \/ 3  + tan(x)   |                   |               ________| |      ___                  |
                cos(x)       3*cos (x)                    \                             sin   (x)/                                                \                \- \/ 3  + tan(x)/                          /                   \             \/ sin(x) / \  - \/ 3  + tan(x)         /
8*sin(2*x) + ------------ + ----------- + -------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------- + -----------------------------------------------------------------------
                 ________        5/2                          /    ___         \                                                                       ___                                                                                         ___                                    
             4*\/ sin(x)    8*sin   (x)                     4*\- \/ 3  + tan(x)/                                                                   - \/ 3  + tan(x)                                                                            - \/ 3  + tan(x)                           
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                         ___                                                                                                                                              
                                                                                                                                     - \/ 3  + tan(x)

$$\frac{\frac{3 \left(4 \sin{\left(2 x \right)} - \frac{\cos{\left(x \right)}}{\sqrt{\sin{\left(x \right)}}}\right) \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - \sqrt{3}}\right) \left(\tan^{2}{\left(x \right)} + 1\right)}{\tan{\left(x \right)} - \sqrt{3}} + 8 \sin{\left(2 x \right)} + \frac{\cos{\left(x \right)}}{4 \sqrt{\sin{\left(x \right)}}} + \frac{3 \cos^{3}{\left(x \right)}}{8 \sin^{\frac{5}{2}}{\left(x \right)}} - \frac{2 \left(\tan^{2}{\left(x \right)} + 1\right) \left(\sqrt{\sin{\left(x \right)}} + \cos{\left(2 x \right)} - 1\right) \left(3 \tan^{2}{\left(x \right)} + 1 - \frac{6 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\tan{\left(x \right)} - \sqrt{3}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)^{2}}{\left(\tan{\left(x \right)} - \sqrt{3}\right)^{2}}\right)}{\tan{\left(x \right)} - \sqrt{3}} + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \left(2 \sqrt{\sin{\left(x \right)}} + 16 \cos{\left(2 x \right)} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}\right)}{4 \left(\tan{\left(x \right)} - \sqrt{3}\right)}}{\tan{\left(x \right)} - \sqrt{3}}$$

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Derivative of (cos2x+sqrtsinx-1)/(tgx-sqrt3)

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The solution