Derivative of pi/sin(x*pi)

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    pi   
---------
sin(x*pi)

$$\frac{\pi}{\sin{\left(\pi x \right)}}$$

pi/sin(x*pi)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \sin{\left(\pi x \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(\pi x \right)}$ :
  1. Let $u = \pi x$ .
  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 $\frac{d}{d x} \pi x$ :
    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: $\pi$
    The result of the chain rule is:
    
    $\pi \cos{\left(\pi x \right)}$
  The result of the chain rule is:
  
  $- \frac{\pi \cos{\left(\pi x \right)}}{\sin^{2}{\left(\pi x \right)}}$
So, the result is: $- \frac{\pi^{2} \cos{\left(\pi x \right)}}{\sin^{2}{\left(\pi x \right)}}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

   2           
-pi *cos(pi*x) 
---------------
      2        
   sin (x*pi)

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

Simplify

The second derivative [src]

    /         2      \
  3 |    2*cos (pi*x)|
pi *|1 + ------------|
    |        2       |
    \     sin (pi*x) /
----------------------
      sin(pi*x)

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

Simplify

The third derivative [src]

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

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

Simplify

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Identical expressions

Derivative of pi/sin(x*pi)

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The solution