Mister Exam

Lang:

The derivative of the function/ 2sin(pix/4)

Derivative of 2sin(pix/4)

The solution

You have entered [src]

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

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

2*sin((pi*x)/4)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi x}{4}$ .
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 $\frac{d}{d x} \frac{\pi x}{4}$ :
  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: $x$ goes to $1$
      So, the result is: $\pi$
    So, the result is: $\frac{\pi}{4}$
  The result of the chain rule is:
  
  $\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{4}$
So, the result is: $\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$
Now simplify:

$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$

The answer is:

$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      /pi*x\
pi*cos|----|
      \ 4  /
------------
     2

$$\frac{\pi \cos{\left(\frac{\pi x}{4} \right)}}{2}$$

Simplify

The second derivative [src]

   2    /pi*x\ 
-pi *sin|----| 
        \ 4  / 
---------------
       8

$$- \frac{\pi^{2} \sin{\left(\frac{\pi x}{4} \right)}}{8}$$

Simplify

The third derivative [src]

   3    /pi*x\ 
-pi *cos|----| 
        \ 4  / 
---------------
       32

$$- \frac{\pi^{3} \cos{\left(\frac{\pi x}{4} \right)}}{32}$$

Simplify