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The derivative of the function/ 1-sin(pix/2)

Derivative of 1-sin(pix/2)

The solution

You have entered [src]

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

$$1 - \sin{\left(\frac{\pi x}{2} \right)}$$

1 - sin((pi*x)/2)

Detail solution

Differentiate $1 - \sin{\left(\frac{\pi x}{2} \right)}$ term by term:
1. The derivative of the constant $1$ is zero.
2. 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 sine is cosine:
    
    $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\pi x}{2}$ :
    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:
    
    $\frac{\pi \cos{\left(\frac{\pi x}{2} \right)}}{2}$
  So, the result is: $- \frac{\pi \cos{\left(\frac{\pi x}{2} \right)}}{2}$
The result is: $- \frac{\pi \cos{\left(\frac{\pi x}{2} \right)}}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  3    /pi*x\
pi *cos|----|
       \ 2  /
-------------
      8

$$\frac{\pi^{3} \cos{\left(\frac{\pi x}{2} \right)}}{8}$$

Simplify