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The derivative of the function/ 0.5sinx-0.25sin(2x)

Derivative of 0.5sinx-0.25sin(2x)

The solution

You have entered [src]

sin(x)   sin(2*x)
------ - --------
  2         4

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

sin(x)/2 - sin(2*x)/4

Detail solution

Differentiate $\frac{\sin{\left(x \right)}}{2} - \frac{\sin{\left(2 x \right)}}{4}$ 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: $\frac{\cos{\left(x \right)}}{2}$
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: $- \frac{\cos{\left(2 x \right)}}{2}$
The result is: $\frac{\cos{\left(x \right)}}{2} - \frac{\cos{\left(2 x \right)}}{2}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify