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The derivative of the function/ 2sinx+2sinxcosx

Derivative of 2sinx+2sinxcosx

The solution

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

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

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

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

-2*(1 + 4*cos(x))*sin(x)

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

Simplify

The third derivative [src]

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

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

Simplify