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Derivative of 3x^2sin(x)

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   2       
3*x *sin(x)

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

(3*x^2)*sin(x)

Detail solution

Apply the product rule:

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

$f{\left(x \right)} = 3 x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  So, the result is: $6 x$
$g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $3 x^{2} \cos{\left(x \right)} + 6 x \sin{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2                    
3*x *cos(x) + 6*x*sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of 3*x^2*sin(x)