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Identical expressions
(two *cos(pi*x/ two))/(five -x)
(2 multiply by co sinus of e of ( Pi multiply by x divide by 2)) divide by (5 minus x)
(two multiply by co sinus of e of ( Pi multiply by x divide by two)) divide by (five minus x)
(2cos(pix/2))/(5-x)
2cospix/2/5-x
(2*cos(pi*x divide by 2)) divide by (5-x)
Similar expressions
(2*cos(pi*x/2))/(5+x)

The derivative of the function/ (2*cos(pi*x/2))/(5-x)

Derivative of (2cos(pix/2))/(5-x)

The solution

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     /pi*x\
2*cos|----|
     \ 2  /
-----------
   5 - x

$$\frac{2 \cos{\left(\frac{\pi x}{2} \right)}}{5 - x}$$

(2*cos((pi*x)/2))/(5 - x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 \cos{\left(\frac{\pi x}{2} \right)}$ and $g{\left(x \right)} = 5 - x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{\pi x}{2}$ .
  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 $\frac{d}{d x} \frac{\pi x}{2}$ :
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. 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: $\pi$
      So, the result is: $\frac{\pi}{2}$
    The result of the chain rule is:
    
    $- \frac{\pi \sin{\left(\frac{\pi x}{2} \right)}}{2}$
  So, the result is: $- \pi \sin{\left(\frac{\pi x}{2} \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $5 - x$ term by term:
  1. The derivative of the constant $5$ is zero.
  2. 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: $-1$
  The result is: $-1$
Now plug in to the quotient rule:

$\frac{- \pi \left(5 - x\right) \sin{\left(\frac{\pi x}{2} \right)} + 2 \cos{\left(\frac{\pi x}{2} \right)}}{\left(5 - x\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     /pi*x\         /pi*x\
2*cos|----|   pi*sin|----|
     \ 2  /         \ 2  /
----------- - ------------
         2       5 - x    
  (5 - x)

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

Simplify

The second derivative [src]

  2    /pi*x\        /pi*x\           /pi*x\
pi *cos|----|   4*cos|----|   2*pi*sin|----|
       \ 2  /        \ 2  /           \ 2  /
------------- - ----------- - --------------
      2                  2        -5 + x    
                 (-5 + x)                   
--------------------------------------------
                   -5 + x

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

Simplify

The third derivative [src]

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

$$\frac{- \frac{\pi^{3} \sin{\left(\frac{\pi x}{2} \right)}}{4} - \frac{3 \pi^{2} \cos{\left(\frac{\pi x}{2} \right)}}{2 \left(x - 5\right)} + \frac{6 \pi \sin{\left(\frac{\pi x}{2} \right)}}{\left(x - 5\right)^{2}} + \frac{12 \cos{\left(\frac{\pi x}{2} \right)}}{\left(x - 5\right)^{3}}}{x - 5}$$

Simplify

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Derivative of (2cos(pix/2))/(5-x)

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Piecewise:

The solution

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Derivative of (2*cos(pi*x/2))/(5-x)

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Derivative of (2cos(pix/2))/(5-x)