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Derivative of:
Derivative of x^-2
Derivative of (-1)/x^2
Derivative of e^x^3
Derivative of (x+1)
Identical expressions
two *x*sin(two *x)- one
2 multiply by x multiply by sinus of (2 multiply by x) minus 1
two multiply by x multiply by sinus of (two multiply by x) minus one
2xsin(2x)-1
2xsin2x-1
Similar expressions
2*x*sin(2*x)+1

The derivative of the function/ 2*x*sin(2*x)-1

Derivative of 2xsin(2*x)-1

The solution

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

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

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

Detail solution

Differentiate $2 x \sin{\left(2 x \right)} - 1$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = 2 x$ ; 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: $x$ goes to $1$
    So, the result is: $2$
  $g{\left(x \right)} = \sin{\left(2 x \right)}$ ; 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: $4 x \cos{\left(2 x \right)} + 2 \sin{\left(2 x \right)}$
2. The derivative of the constant $-1$ is zero.
The result is: $4 x \cos{\left(2 x \right)} + 2 \sin{\left(2 x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

2*sin(2*x) + 4*x*cos(2*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

-8*(3*sin(2*x) + 2*x*cos(2*x))

$$- 8 \left(2 x \cos{\left(2 x \right)} + 3 \sin{\left(2 x \right)}\right)$$

Simplify