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Identical expressions
((five -x)*cos((pi*x)/ two))
((5 minus x) multiply by co sinus of e of (( Pi multiply by x) divide by 2))
((five minus x) multiply by co sinus of e of (( Pi multiply by x) divide by two))
((5-x)cos((pix)/2))
5-xcospix/2
((5-x)*cos((pi*x) divide by 2))
Similar expressions
((5+x)*cos((pi*x)/2))

The derivative of the function/ ((5-x)*cos((pi*x)/2))

Derivative of ((5-x)cos((pix)/2))

The solution

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           /pi*x\
(5 - x)*cos|----|
           \ 2  /

$$\left(5 - x\right) \cos{\left(\frac{\pi x}{2} \right)}$$

(5 - x)*cos((pi*x)/2)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = 5 - x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $5 - x$ term by term:
  1. The derivative of the constant $5$ is zero.
  2. 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: $-1$
  The result is: $-1$
$g{\left(x \right)} = \cos{\left(\frac{\pi x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\pi x}{2}$ .
2. The derivative of cosine is negative sine:
  
  $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
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.
      1. 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:
  
  $- \frac{\pi \sin{\left(\frac{\pi x}{2} \right)}}{2}$
The result is: $- \frac{\pi \left(5 - x\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}$
Now simplify:

$\frac{\pi \left(x - 5\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}$

The answer is:

$\frac{\pi \left(x - 5\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}$

The graph

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The first derivative [src]

                            /pi*x\
              pi*(5 - x)*sin|----|
     /pi*x\                 \ 2  /
- cos|----| - --------------------
     \ 2  /            2

$$- \frac{\pi \left(5 - x\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}$$

Simplify

The second derivative [src]

   /               /pi*x\            \
   |pi*(-5 + x)*cos|----|            |
   |               \ 2  /      /pi*x\|
pi*|--------------------- + sin|----||
   \          4                \ 2  //

$$\pi \left(\frac{\pi \left(x - 5\right) \cos{\left(\frac{\pi x}{2} \right)}}{4} + \sin{\left(\frac{\pi x}{2} \right)}\right)$$

Simplify

The third derivative [src]

  2 /     /pi*x\                  /pi*x\\
pi *|6*cos|----| - pi*(-5 + x)*sin|----||
    \     \ 2  /                  \ 2  //
-----------------------------------------
                    8

$$\frac{\pi^{2} \left(- \pi \left(x - 5\right) \sin{\left(\frac{\pi x}{2} \right)} + 6 \cos{\left(\frac{\pi x}{2} \right)}\right)}{8}$$

Simplify

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Derivative of ((5-x)cos((pix)/2))

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The solution

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Derivative of ((5-x)*cos((pi*x)/2))

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Piecewise:

The solution

Derivative of ((5-x)cos((pix)/2))