Derivative of x^2sin2x

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 2         
x *sin(2*x)

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

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

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

Transform

Detail solution

Apply the product rule:

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

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

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

The answer is:

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

The graph

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The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                               2         \
4*\3*cos(2*x) - 6*x*sin(2*x) - 2*x *cos(2*x)/

$$4 \left(- 2 x^{2} \cos{\left(2 x \right)} - 6 x \sin{\left(2 x \right)} + 3 \cos{\left(2 x \right)}\right)$$

Simplify

The graph