You entered:

y=sin2xcosx/2

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

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

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

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

Derivative of y=sin2xcosx/2

You have entered [src]

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

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

d /sin(2*x)*cos(x)\
--|---------------|
dx\       2       /

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

Transform

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of cosine is negative sine:
    
    $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
  $g{\left(x \right)} = \sin{\left(2 x \right)}$ ; 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: $- \sin{\left(x \right)} \sin{\left(2 x \right)} + 2 \cos{\left(x \right)} \cos{\left(2 x \right)}$
So, the result is: $- \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{2} + \cos{\left(x \right)} \cos{\left(2 x \right)}$
Now simplify:

$\frac{\cos{\left(x \right)}}{4} + \frac{3 \cos{\left(3 x \right)}}{4}$

The answer is:

$\frac{\cos{\left(x \right)}}{4} + \frac{3 \cos{\left(3 x \right)}}{4}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

 /                    5*cos(x)*sin(2*x)\
-|2*cos(2*x)*sin(x) + -----------------|
 \                            2        /

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

Simplify

The third derivative [src]

                     13*sin(x)*sin(2*x)
-7*cos(x)*cos(2*x) + ------------------
                             2

$$\frac{13 \sin{\left(x \right)} \sin{\left(2 x \right)}}{2} - 7 \cos{\left(x \right)} \cos{\left(2 x \right)}$$

Simplify

The graph