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Derivative of sin(pi*x/4)

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   /pi*x\
sin|----|
   \ 4  /

$$\sin{\left(\frac{\pi x}{4} \right)}$$

d /   /pi*x\\
--|sin|----||
dx\   \ 4  //

$$\frac{d}{d x} \sin{\left(\frac{\pi x}{4} \right)}$$

Transform

Detail solution

Let $u = \frac{\pi x}{4}$ .
The derivative of sine is cosine:

$\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\pi x}{4}$ :
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: $\frac{\pi}{4}$
The result of the chain rule is:

$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{4}$

The answer is:

$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      /pi*x\
pi*cos|----|
      \ 4  /
------------
     4

$$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{4}$$

Simplify

The second derivative [src]

   2    /pi*x\ 
-pi *sin|----| 
        \ 4  / 
---------------
       16

$$- \frac{\pi^{2} \sin{\left(\frac{\pi x}{4} \right)}}{16}$$

Simplify

The third derivative [src]

   3    /pi*x\ 
-pi *cos|----| 
        \ 4  / 
---------------
       64

$$- \frac{\pi^{3} \cos{\left(\frac{\pi x}{4} \right)}}{64}$$

Simplify

The graph