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

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Derivative of:
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Identical expressions
one -x/ one -sin(pi*x)/ two
1 minus x divide by 1 minus sinus of ( Pi multiply by x) divide by 2
one minus x divide by one minus sinus of ( Pi multiply by x) divide by two
1-x/1-sin(pix)/2
1-x/1-sinpix/2
1-x divide by 1-sin(pi*x) divide by 2
Similar expressions
1-x/1+sin(pi*x)/2
1+x/1-sin(pi*x)/2

The derivative of the function/ 1-x/1-sin(pi*x)/2

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

The solution

You have entered [src]

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

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

d /    x   sin(pi*x)\
--|1 - - - ---------|
dx\    1       2    /

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

Transform

Detail solution

Differentiate $- \frac{x}{1} - \frac{\sin{\left(\pi x \right)}}{2} + 1$ 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. 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: $1^{-1}$
  So, the result is: $-1$
3. 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. 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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $\pi$
      The result of the chain rule is:
      
      $\pi \cos{\left(\pi x \right)}$
    So, the result is: $\frac{\pi \cos{\left(\pi x \right)}}{2}$
  So, the result is: $- \frac{\pi \cos{\left(\pi x \right)}}{2}$
The result is: $- \frac{\pi \cos{\left(\pi x \right)}}{2} - 1$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

     pi*cos(pi*x)
-1 - ------------
          2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  3          
pi *cos(pi*x)
-------------
      2

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

Simplify

The graph

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