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Derivative of:
Derivative of (x+2)
Derivative of 1+x
Derivative of (sqrt(x)-1)/sqrt(x^2-x-1)
Derivative of e^x*cos(x)
Identical expressions
three *sin(pi*x/ twelve)
3 multiply by sinus of ( Pi multiply by x divide by 12)
three multiply by sinus of ( Pi multiply by x divide by twelve)
3sin(pix/12)
3sinpix/12
3*sin(pi*x divide by 12)

The derivative of the function/ 3*sin(pi*x/12)

Derivative of 3sin(pix/12)

The solution

You have entered [src]

     /pi*x\
3*sin|----|
     \ 12 /

$$3 \sin{\left(\frac{\pi x}{12} \right)}$$

3*sin((pi*x)/12)

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi x}{12}$ .
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 $\frac{d}{d x} \frac{\pi x}{12}$ :
  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: $\frac{\pi}{12}$
  The result of the chain rule is:
  
  $\frac{\pi \cos{\left(\frac{\pi x}{12} \right)}}{12}$
So, the result is: $\frac{\pi \cos{\left(\frac{\pi x}{12} \right)}}{4}$
Now simplify:

$\frac{\pi \cos{\left(\frac{\pi x}{12} \right)}}{4}$

The answer is:

$\frac{\pi \cos{\left(\frac{\pi x}{12} \right)}}{4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      /pi*x\
pi*cos|----|
      \ 12 /
------------
     4

$$\frac{\pi \cos{\left(\frac{\pi x}{12} \right)}}{4}$$

Simplify

The second derivative [src]

   2    /pi*x\ 
-pi *sin|----| 
        \ 12 / 
---------------
       48

$$- \frac{\pi^{2} \sin{\left(\frac{\pi x}{12} \right)}}{48}$$

Simplify

The third derivative [src]

   3    /pi*x\ 
-pi *cos|----| 
        \ 12 / 
---------------
      576

$$- \frac{\pi^{3} \cos{\left(\frac{\pi x}{12} \right)}}{576}$$

Simplify