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Derivative of:
Derivative of 3*x^2-1/x^3
Derivative of 3^(1/x)
Derivative of (2*x-1)^2
Derivative of 2*x^3-3*x^2+6*x+1
Identical expressions
(two / three)*sin(3x/ two)
(2 divide by 3) multiply by sinus of (3x divide by 2)
(two divide by three) multiply by sinus of (3x divide by two)
(2/3)sin(3x/2)
2/3sin3x/2
(2 divide by 3)*sin(3x divide by 2)

The derivative of the function/ (2/3)*sin(3x/2)

Derivative of (2/3)*sin(3x/2)

The solution

You have entered [src]

     /3*x\
2*sin|---|
     \ 2 /
----------
    3

$$\frac{2 \sin{\left(\frac{3 x}{2} \right)}}{3}$$

2*sin((3*x)/2)/3

Detail solution

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

$\cos{\left(\frac{3 x}{2} \right)}$

The answer is:

$\cos{\left(\frac{3 x}{2} \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   /3*x\
cos|---|
   \ 2 /

$$\cos{\left(\frac{3 x}{2} \right)}$$

Simplify

The second derivative [src]

      /3*x\
-3*sin|---|
      \ 2 /
-----------
     2

$$- \frac{3 \sin{\left(\frac{3 x}{2} \right)}}{2}$$

Simplify

The third derivative [src]

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

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

Simplify