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The derivative of the function/ sinx+(sin2x/2)

Derivative of sinx+(sin2x/2)

The solution

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

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

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

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

Transform

Detail solution

Differentiate $\sin{\left(x \right)} + \frac{\sin{\left(2 x \right)}}{2}$ term by term:
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  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:
    
    $2 \cos{\left(2 x \right)}$
  So, the result is: $\cos{\left(2 x \right)}$
The result is: $\cos{\left(x \right)} + \cos{\left(2 x \right)}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

cos(x) + cos(2*x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Derivative of sinx+(sin2x/2)