Derivative of x^3sinxcosx

The solution

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 3              
x *sin(x)*cos(x)

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

(x^3*sin(x))*cos(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x^{3} \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{3}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}$
$g{\left(x \right)} = \cos{\left(x \right)}$ ; 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: $- x^{3} \sin^{2}{\left(x \right)} + \left(x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)}\right) \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

  //            2                    \           2                                                 \
x*\\6*sin(x) - x *sin(x) + 6*x*cos(x)/*cos(x) - x *cos(x)*sin(x) - 2*x*(3*sin(x) + x*cos(x))*sin(x)/

$$x \left(- x^{2} \sin{\left(x \right)} \cos{\left(x \right)} - 2 x \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \sin{\left(x \right)} + \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) \cos{\left(x \right)}\right)$$

Simplify

The third derivative [src]

 3    2      /             3                           2       \              /            2                    \             2                             
x *sin (x) - \-6*sin(x) + x *cos(x) - 18*x*cos(x) + 9*x *sin(x)/*cos(x) - 3*x*\6*sin(x) - x *sin(x) + 6*x*cos(x)/*sin(x) - 3*x *(3*sin(x) + x*cos(x))*cos(x)

$$x^{3} \sin^{2}{\left(x \right)} - 3 x^{2} \left(x \cos{\left(x \right)} + 3 \sin{\left(x \right)}\right) \cos{\left(x \right)} - 3 x \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) \sin{\left(x \right)} - \left(x^{3} \cos{\left(x \right)} + 9 x^{2} \sin{\left(x \right)} - 18 x \cos{\left(x \right)} - 6 \sin{\left(x \right)}\right) \cos{\left(x \right)}$$

Simplify

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Derivative of x^3sinxcosx

The graph:

Piecewise:

The solution