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Derivative of x-lnx
Derivative of (x^2-8*x)/(x+1)
Derivative of x^2+5
Derivative of x^2-4*x
Identical expressions
(4П-cosx)^(one / two)+cosx/(4П*sinx^ two)
(4П minus co sinus of e of x) to the power of (1 divide by 2) plus co sinus of e of x divide by (4П multiply by sinus of x squared )
(4П minus co sinus of e of x) to the power of (one divide by two) plus co sinus of e of x divide by (4П multiply by sinus of x to the power of two)
(4П-cosx)(1/2)+cosx/(4П*sinx2)
4П-cosx1/2+cosx/4П*sinx2
(4П-cosx)^(1/2)+cosx/(4П*sinx²)
(4П-cosx) to the power of (1/2)+cosx/(4П*sinx to the power of 2)
(4П-cosx)^(1/2)+cosx/(4Пsinx^2)
(4П-cosx)(1/2)+cosx/(4Пsinx2)
4П-cosx1/2+cosx/4Пsinx2
4П-cosx^1/2+cosx/4Пsinx^2
(4П-cosx)^(1 divide by 2)+cosx divide by (4П*sinx^2)
Similar expressions
(4П+cosx)^(1/2)+cosx/(4П*sinx^2)
(4П-cosx)^(1/2)-cosx/(4П*sinx^2)

The derivative of the function/ (4П-cosx)^(1/2)+cosx/(4П*sinx^2)

Derivative of (4П-cosx)^(1/2)+cosx/(4П*sinx^2)

The solution

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  _______________      cos(x)   
\/ 4*pi - cos(x)  + ------------
                            2   
                    4*pi*sin (x)

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

sqrt(4*pi - cos(x)) + cos(x)/(((4*pi)*sin(x)^2))

Detail solution

Differentiate $\sqrt{- \cos{\left(x \right)} + 4 \pi} + \frac{\cos{\left(x \right)}}{4 \pi \sin^{2}{\left(x \right)}}$ term by term:
1. Let $u = - \cos{\left(x \right)} + 4 \pi$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \cos{\left(x \right)} + 4 \pi\right)$ :
  1. Differentiate $- \cos{\left(x \right)} + 4 \pi$ term by term:
    1. The derivative of the constant $4 \pi$ is zero.
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      So, the result is: $\sin{\left(x \right)}$
    The result is: $\sin{\left(x \right)}$
  The result of the chain rule is:
  
  $\frac{\sin{\left(x \right)}}{2 \sqrt{- \cos{\left(x \right)} + 4 \pi}}$
4. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = 4 \pi \sin^{2}{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \sin{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. 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:
      
      $2 \sin{\left(x \right)} \cos{\left(x \right)}$
    So, the result is: $8 \pi \sin{\left(x \right)} \cos{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- 4 \pi \sin^{3}{\left(x \right)} - 8 \pi \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{16 \pi^{2} \sin^{4}{\left(x \right)}}$
The result is: $\frac{- 4 \pi \sin^{3}{\left(x \right)} - 8 \pi \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{16 \pi^{2} \sin^{4}{\left(x \right)}} + \frac{\sin{\left(x \right)}}{2 \sqrt{- \cos{\left(x \right)} + 4 \pi}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                                                 2      
       sin(x)              1                  cos (x)   
------------------- - ------------*sin(x) - ------------
    _______________           2                     3   
2*\/ 4*pi - cos(x)    4*pi*sin (x)          2*pi*sin (x)

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

Simplify

The second derivative [src]

           2                                                   3    
        sin (x)              2*cos(x)         5*cos(x)    6*cos (x) 
- ------------------- + ------------------ + ---------- + ----------
                  3/2     ________________         2            4   
  (-cos(x) + 4*pi)      \/ -cos(x) + 4*pi    pi*sin (x)   pi*sin (x)
--------------------------------------------------------------------
                                 4

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

Simplify

The third derivative [src]

                                                  3                 2            4                            
       5               sin(x)                3*sin (x)         7*cos (x)    6*cos (x)       3*cos(x)*sin(x)   
- ----------- - -------------------- + --------------------- - ---------- - ---------- - ---------------------
  4*pi*sin(x)       ________________                     5/2         3            5                        3/2
                2*\/ -cos(x) + 4*pi    8*(-cos(x) + 4*pi)      pi*sin (x)   pi*sin (x)   4*(-cos(x) + 4*pi)

$$- \frac{5}{4 \pi \sin{\left(x \right)}} - \frac{7 \cos^{2}{\left(x \right)}}{\pi \sin^{3}{\left(x \right)}} - \frac{6 \cos^{4}{\left(x \right)}}{\pi \sin^{5}{\left(x \right)}} - \frac{\sin{\left(x \right)}}{2 \sqrt{- \cos{\left(x \right)} + 4 \pi}} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{4 \left(- \cos{\left(x \right)} + 4 \pi\right)^{\frac{3}{2}}} + \frac{3 \sin^{3}{\left(x \right)}}{8 \left(- \cos{\left(x \right)} + 4 \pi\right)^{\frac{5}{2}}}$$

Simplify

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Derivative of (4П-cosx)^(1/2)+cosx/(4П*sinx^2)

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The solution