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

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

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The solution

You have entered [src] 
           /pi*x\
(5 - x)*cos|----|
           \ 2  /
(5−x)cos⁡(πx2)\left(5 - x\right) \cos{\left(\frac{\pi x}{2} \right)} 
(5 - x)*cos((pi*x)/2)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=5−xf{\left(x \right)} = 5 - x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Differentiate 5−x5 - x term by term:

      1. The derivative of the constant 55 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: xx goes to 11

        So, the result is: −1-1

      The result is: −1-1

    g(x)=cos⁡(πx2)g{\left(x \right)} = \cos{\left(\frac{\pi x}{2} \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=πx2u = \frac{\pi x}{2}.

    2. The derivative of cosine is negative sine:

      dducos⁡(u)=−sin⁡(u)\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}

    3. Then, apply the chain rule. Multiply by ddxπx2\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: xx goes to 11

          So, the result is: π\pi

        So, the result is: π2\frac{\pi}{2}

      The result of the chain rule is:

      −πsin⁡(πx2)2- \frac{\pi \sin{\left(\frac{\pi x}{2} \right)}}{2}

    The result is: −π(5−x)sin⁡(πx2)2−cos⁡(πx2)- \frac{\pi \left(5 - x\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}

  2. Now simplify:

    π(x−5)sin⁡(πx2)2−cos⁡(πx2)\frac{\pi \left(x - 5\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}

The answer is:

π(x−5)sin⁡(πx2)2−cos⁡(πx2)\frac{\pi \left(x - 5\right) \sin{\left(\frac{\pi x}{2} \right)}}{2} - \cos{\left(\frac{\pi x}{2} \right)}

The graph
02468-8-6-4-2-1010-5050
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The first derivative [src] 
                            /pi*x\
              pi*(5 - x)*sin|----|
     /pi*x\                 \ 2  /
- cos|----| - --------------------
     \ 2  /            2
−π(5−x)sin⁡(πx2)2−cos⁡(πx2)- \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  //
π(π(x−5)cos⁡(πx2)4+sin⁡(πx2))\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
π2(−π(x−5)sin⁡(πx2)+6cos⁡(πx2))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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