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Derivative of:
Derivative of (sin(x))^2
Derivative of x-1
Derivative of sin(x)/x^2
Derivative of cos(3x)
Identical expressions
4pi/ three *sin(pi*x/ three)
4 Pi divide by 3 multiply by sinus of ( Pi multiply by x divide by 3)
4 Pi divide by three multiply by sinus of ( Pi multiply by x divide by three)
4pi/3sin(pix/3)
4pi/3sinpix/3
4pi divide by 3*sin(pi*x divide by 3)

The derivative of the function/ 4pi/3*sin(pi*x/3)

Derivative of 4pi/3sin(pix/3)

The solution

You have entered [src]

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

$$\frac{4 \pi}{3} \sin{\left(\frac{\pi x}{3} \right)}$$

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = \frac{\pi x}{3}$ .
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 $\frac{d}{d x} \frac{\pi x}{3}$ :
  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: $\frac{\pi}{3}$
  The result of the chain rule is:
  
  $\frac{\pi \cos{\left(\frac{\pi x}{3} \right)}}{3}$
So, the result is: $\frac{4 \pi^{2} \cos{\left(\frac{\pi x}{3} \right)}}{9}$
Now simplify:

$\frac{4 \pi^{2} \cos{\left(\frac{\pi x}{3} \right)}}{9}$

The answer is:

$\frac{4 \pi^{2} \cos{\left(\frac{\pi x}{3} \right)}}{9}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    2    /pi*x\
4*pi *cos|----|
         \ 3  /
---------------
       9

$$\frac{4 \pi^{2} \cos{\left(\frac{\pi x}{3} \right)}}{9}$$

Simplify

The second derivative [src]

     3    /pi*x\
-4*pi *sin|----|
          \ 3  /
----------------
       27

$$- \frac{4 \pi^{3} \sin{\left(\frac{\pi x}{3} \right)}}{27}$$

Simplify

The third derivative [src]

     4    /pi*x\
-4*pi *cos|----|
          \ 3  /
----------------
       81

$$- \frac{4 \pi^{4} \cos{\left(\frac{\pi x}{3} \right)}}{81}$$

Simplify