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Similar expressions
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The derivative of the function/ 0,5-cos(cos(y)+2)

Derivative of 0,5-cos(cos(y)+2)

The solution

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1/2 - cos(cos(y) + 2)

$$\frac{1}{2} - \cos{\left(\cos{\left(y \right)} + 2 \right)}$$

1/2 - cos(cos(y) + 2)

Detail solution

Differentiate $\frac{1}{2} - \cos{\left(\cos{\left(y \right)} + 2 \right)}$ term by term:
1. The derivative of the constant $\frac{1}{2}$ is zero.
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \cos{\left(y \right)} + 2$ .
  2. The derivative of cosine is negative sine:
    
    $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d y} \left(\cos{\left(y \right)} + 2\right)$ :
    1. Differentiate $\cos{\left(y \right)} + 2$ term by term:
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d y} \cos{\left(y \right)} = - \sin{\left(y \right)}$
      2. The derivative of the constant $2$ is zero.
      The result is: $- \sin{\left(y \right)}$
    The result of the chain rule is:
    
    $\sin{\left(y \right)} \sin{\left(\cos{\left(y \right)} + 2 \right)}$
  So, the result is: $- \sin{\left(y \right)} \sin{\left(\cos{\left(y \right)} + 2 \right)}$
The result is: $- \sin{\left(y \right)} \sin{\left(\cos{\left(y \right)} + 2 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/   2                                                                \       
\sin (y)*sin(2 + cos(y)) + 3*cos(y)*cos(2 + cos(y)) + sin(2 + cos(y))/*sin(y)

$$\left(\sin^{2}{\left(y \right)} \sin{\left(\cos{\left(y \right)} + 2 \right)} + \sin{\left(\cos{\left(y \right)} + 2 \right)} + 3 \cos{\left(y \right)} \cos{\left(\cos{\left(y \right)} + 2 \right)}\right) \sin{\left(y \right)}$$

Simplify

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Derivative of 0,5-cos(cos(y)+2)

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The solution