Derivative of x^2sin3x

You have entered [src]

 2         
x *sin(3*x)

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

d / 2         \
--\x *sin(3*x)/
dx

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

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 $2 x$
$g{\left(x \right)} = \sin{\left(3 x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = 3 x$ .
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 $\frac{d}{d x} 3 x$ :
  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: $3$
  The result of the chain rule is:
  
  $3 \cos{\left(3 x \right)}$
The result is: $3 x^{2} \cos{\left(3 x \right)} + 2 x \sin{\left(3 x \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                2                         
2*sin(3*x) - 9*x *sin(3*x) + 12*x*cos(3*x)

$$- 9 x^{2} \sin{\left(3 x \right)} + 12 x \cos{\left(3 x \right)} + 2 \sin{\left(3 x \right)}$$

Simplify

The third derivative [src]

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

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

Simplify

The graph