Derivative of xsin^2x

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     2   
x*sin (x)

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

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

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

Transform

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
$g{\left(x \right)} = \sin^{2}{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = \sin{\left(x \right)}$ .
2. Apply the power rule: $u^{2}$ goes to $2 u$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $2 \sin{\left(x \right)} \cos{\left(x \right)}$
The result is: $2 x \sin{\left(x \right)} \cos{\left(x \right)} + \sin^{2}{\left(x \right)}$
Now simplify:

$x \sin{\left(2 x \right)} - \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$

The answer is:

$x \sin{\left(2 x \right)} - \frac{\cos{\left(2 x \right)}}{2} + \frac{1}{2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /       2           2                       \
2*\- 3*sin (x) + 3*cos (x) - 4*x*cos(x)*sin(x)/

$$2 \left(- 4 x \sin{\left(x \right)} \cos{\left(x \right)} - 3 \sin^{2}{\left(x \right)} + 3 \cos^{2}{\left(x \right)}\right)$$

Simplify

The graph