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The derivative of the function/ 1/3x^3*sin2x

Derivative of 1/3x^3*sin2x

The solution

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 3         
x *sin(2*x)
-----------
     3

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

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

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

Transform

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of 1/3x^3*sin2x

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The solution