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Derivative of cos(piy/(2b))

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   /pi*y\
cos|----|
   \2*b /

$$\cos{\left(\frac{\pi y}{2 b} \right)}$$

cos((pi*y)/((2*b)))

Detail solution

Let $u = \frac{\pi y}{2 b}$ .
The derivative of cosine is negative sine:

$\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{\partial}{\partial y} \frac{\pi y}{2 b}$ :
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.
    1. Apply the power rule: $y$ goes to $1$
    So, the result is: $\pi$
  So, the result is: $\pi \frac{1}{2 b}$
The result of the chain rule is:

$- \frac{\pi \sin{\left(\frac{\pi y}{2 b} \right)}}{2 b}$
Now simplify:

$- \frac{\pi \sin{\left(\frac{\pi y}{2 b} \right)}}{2 b}$

The answer is:

$- \frac{\pi \sin{\left(\frac{\pi y}{2 b} \right)}}{2 b}$

The first derivative [src]

       /pi*y\ 
-pi*sin|----| 
       \2*b / 
--------------
     2*b

$$- \frac{\pi \sin{\left(\frac{\pi y}{2 b} \right)}}{2 b}$$

Simplify

The second derivative [src]

   2    /pi*y\ 
-pi *cos|----| 
        \2*b / 
---------------
         2     
      4*b

$$- \frac{\pi^{2} \cos{\left(\frac{\pi y}{2 b} \right)}}{4 b^{2}}$$

Simplify

4-я производная [src]

  4    /pi*y\
pi *cos|----|
       \2*b /
-------------
        4    
    16*b

$$\frac{\pi^{4} \cos{\left(\frac{\pi y}{2 b} \right)}}{16 b^{4}}$$

Simplify

The third derivative [src]

  3    /pi*y\
pi *sin|----|
       \2*b /
-------------
        3    
     8*b

$$\frac{\pi^{3} \sin{\left(\frac{\pi y}{2 b} \right)}}{8 b^{3}}$$

Simplify

Derivative of cos(pi*y/(2*b))