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Identical expressions
5e^(x/ two)*sin(pix/ two)
5e to the power of (x divide by 2) multiply by sinus of ( Pi x divide by 2)
5e to the power of (x divide by two) multiply by sinus of ( Pi x divide by two)
5e(x/2)*sin(pix/2)
5ex/2*sinpix/2
5e^(x/2)sin(pix/2)
5e(x/2)sin(pix/2)
5ex/2sinpix/2
5e^x/2sinpix/2
5e^(x divide by 2)*sin(pix divide by 2)

The derivative of the function/ 5e^(x/2)*sin(pix/2)

Derivative of 5e^(x/2)*sin(pix/2)

The solution

You have entered [src]

   x          
   -          
   2    /pi*x\
5*E *sin|----|
        \ 2  /

$$5 e^{\frac{x}{2}} \sin{\left(\frac{\pi x}{2} \right)}$$

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 5 e^{\frac{x}{2}}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \frac{x}{2}$ .
  2. The derivative of $e^{u}$ is itself.
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x}{2}$ :
    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: $\frac{1}{2}$
    The result of the chain rule is:
    
    $\frac{e^{\frac{x}{2}}}{2}$
  So, the result is: $\frac{5 e^{\frac{x}{2}}}{2}$
$g{\left(x \right)} = \sin{\left(\frac{\pi x}{2} \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \frac{\pi x}{2}$ .
2. The derivative of 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.
      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:
  
  $\frac{\pi \cos{\left(\frac{\pi x}{2} \right)}}{2}$
The result is: $\frac{5 e^{\frac{x}{2}} \sin{\left(\frac{\pi x}{2} \right)}}{2} + \frac{5 \pi e^{\frac{x}{2}} \cos{\left(\frac{\pi x}{2} \right)}}{2}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   x                             x
   -                             -
   2    /pi*x\           /pi*x\  2
5*e *sin|----|   5*pi*cos|----|*e 
        \ 2  /           \ 2  /   
-------------- + -----------------
      2                  2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

                                                                    x
                                                                    -
  /    3    /pi*x\       2    /pi*x\           /pi*x\      /pi*x\\  2
5*|- pi *cos|----| - 3*pi *sin|----| + 3*pi*cos|----| + sin|----||*e 
  \         \ 2  /            \ 2  /           \ 2  /      \ 2  //   
---------------------------------------------------------------------
                                  8

$$\frac{5 \left(- 3 \pi^{2} \sin{\left(\frac{\pi x}{2} \right)} + \sin{\left(\frac{\pi x}{2} \right)} - \pi^{3} \cos{\left(\frac{\pi x}{2} \right)} + 3 \pi \cos{\left(\frac{\pi x}{2} \right)}\right) e^{\frac{x}{2}}}{8}$$

Simplify

The graph

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Identical expressions

Derivative of 5e^(x/2)*sin(pix/2)

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The solution