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Derivative of:
Derivative of sin(4x)
Derivative of (x^3-1)(x^2+x+1)
Derivative of ex^2
Derivative of x^3/(2*x+4)
Identical expressions
(x*sin(two *x))/ two
(x multiply by sinus of (2 multiply by x)) divide by 2
(x multiply by sinus of (two multiply by x)) divide by two
(xsin(2x))/2
xsin2x/2
(x*sin(2*x)) divide by 2
Similar expressions
5*arcsin(sqrt(x))-ln(ln(x))/5^x+x*sin^2(x)/2-x^3

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

Derivative of (xsin(2x))/2

The solution

You have entered [src]

x*sin(2*x)
----------
    2

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \sin{\left(2 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = 2 x$ .
  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 $\frac{d}{d x} 2 x$ :
    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: $2$
    The result of the chain rule is:
    
    $2 \cos{\left(2 x \right)}$
  The result is: $2 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}$
So, the result is: $x \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{2}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

sin(2*x)             
-------- + x*cos(2*x)
   2

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

Simplify

The second derivative [src]

2*(-x*sin(2*x) + cos(2*x))

$$2 \left(- x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right)$$

Simplify

The third derivative [src]

-2*(3*sin(2*x) + 2*x*cos(2*x))

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

Simplify