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The derivative of the function/ sqrt(cos(x)-1)

Derivative of sqrt(cos(x)-1)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  ____________
\/ cos(x) - 1
$$\sqrt{\cos{\left(x \right)} - 1}$$ 
sqrt(cos(x) - 1)
Detail solution

  1. Let .

  2. Apply the power rule: goes to

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. The derivative of cosine is negative sine:

      2. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    -sin(x)     
----------------
    ____________
2*\/ cos(x) - 1
$$- \frac{\sin{\left(x \right)}}{2 \sqrt{\cos{\left(x \right)} - 1}}$$
Simplify
The second derivative [src] 
 /                2     \ 
 |             sin (x)  | 
-|2*cos(x) + -----------| 
 \           -1 + cos(x)/ 
--------------------------
        _____________     
    4*\/ -1 + cos(x)
$$- \frac{2 \cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}}{4 \sqrt{\cos{\left(x \right)} - 1}}$$
Simplify
3-я производная [src] 
/                         2      \       
|      6*cos(x)      3*sin (x)   |       
|4 - ----------- - --------------|*sin(x)
|    -1 + cos(x)                2|       
\                  (-1 + cos(x)) /       
-----------------------------------------
                _____________            
            8*\/ -1 + cos(x)
$$\frac{\left(4 - \frac{6 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{3 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \sin{\left(x \right)}}{8 \sqrt{\cos{\left(x \right)} - 1}}$$
Simplify
The third derivative [src] 
/                         2      \       
|      6*cos(x)      3*sin (x)   |       
|4 - ----------- - --------------|*sin(x)
|    -1 + cos(x)                2|       
\                  (-1 + cos(x)) /       
-----------------------------------------
                _____________            
            8*\/ -1 + cos(x)
$$\frac{\left(4 - \frac{6 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{3 \sin^{2}{\left(x \right)}}{\left(\cos{\left(x \right)} - 1\right)^{2}}\right) \sin{\left(x \right)}}{8 \sqrt{\cos{\left(x \right)} - 1}}$$
Simplify
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