Derivative of xsinxcosx

The solution

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x*sin(x)*cos(x)

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

(x*sin(x))*cos(x)

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

$f{\left(x \right)} = x \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
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 $1$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $x \cos{\left(x \right)} + \sin{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
The result is: $- x \sin^{2}{\left(x \right)} + \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \cos{\left(x \right)}$
Now simplify:

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

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]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of xsinxcosx

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The solution