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Derivative of x^2*sin(x)

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 2       
x *sin(x)

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

x^2*sin(x)

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

            2                    
2*sin(x) - x *sin(x) + 4*x*cos(x)

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

Simplify

The third derivative [src]

            2                    
6*cos(x) - x *cos(x) - 6*x*sin(x)

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

Simplify

The graph