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Derivative of f(x)=2sinx(2x²-1)

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         /   2    \
2*sin(x)*\2*x  - 1/

$$2 \cdot \left(2 x^{2} - 1\right) \sin{\left(x \right)}$$

d /         /   2    \\
--\2*sin(x)*\2*x  - 1//
dx

$$\frac{d}{d x} 2 \cdot \left(2 x^{2} - 1\right) \sin{\left(x \right)}$$

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:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 2 x^{2} - 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $2 x^{2} - 1$ term by term:
    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: $4 x$
    2. The derivative of the constant $\left(-1\right) 1$ is zero.
    The result is: $4 x$
  $g{\left(x \right)} = \sin{\left(x \right)}$ ; 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: $4 x \sin{\left(x \right)} + \left(2 x^{2} - 1\right) \cos{\left(x \right)}$
So, the result is: $8 x \sin{\left(x \right)} + 2 \cdot \left(2 x^{2} - 1\right) \cos{\left(x \right)}$
Now simplify:

$8 x \sin{\left(x \right)} + 2 \cdot \left(2 x^{2} - 1\right) \cos{\left(x \right)}$

The answer is:

$8 x \sin{\left(x \right)} + 2 \cdot \left(2 x^{2} - 1\right) \cos{\left(x \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  /   2    \                    
2*\2*x  - 1/*cos(x) + 8*x*sin(x)

$$8 x \sin{\left(x \right)} + 2 \cdot \left(2 x^{2} - 1\right) \cos{\left(x \right)}$$

Simplify

The second derivative [src]

  /           /        2\                    \
2*\4*sin(x) - \-1 + 2*x /*sin(x) + 8*x*cos(x)/

$$2 \cdot \left(8 x \cos{\left(x \right)} - \left(2 x^{2} - 1\right) \sin{\left(x \right)} + 4 \sin{\left(x \right)}\right)$$

Simplify

The third derivative [src]

  /            /        2\                     \
2*\12*cos(x) - \-1 + 2*x /*cos(x) - 12*x*sin(x)/

$$2 \left(- 12 x \sin{\left(x \right)} - \left(2 x^{2} - 1\right) \cos{\left(x \right)} + 12 \cos{\left(x \right)}\right)$$

Simplify

The graph