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The derivative of the function/ 2(-xcosx+sinx)

Derivative of 2(-xcosx+sinx)

The solution

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Differentiate $- x \cos{\left(x \right)} + \sin{\left(x \right)}$ term by term:
  1. Apply the product rule:
    
    $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
    
    $f{\left(x \right)} = - x$ ; 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: $-1$
    $g{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
    1. The derivative of cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    The result is: $x \sin{\left(x \right)} - \cos{\left(x \right)}$
  2. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $x \sin{\left(x \right)}$
So, the result is: $2 x \sin{\left(x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

2*x*sin(x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify