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Similar expressions
(2cosx-1)^(1/2)

The derivative of the function/ (2cosx+1)^(1/2)

Derivative of (2cosx+1)^(1/2)

The solution

You have entered [src]

  ______________
\/ 2*cos(x) + 1

$$\sqrt{2 \cos{\left(x \right)} + 1}$$

d /  ______________\
--\\/ 2*cos(x) + 1 /
dx

$$\frac{d}{d x} \sqrt{2 \cos{\left(x \right)} + 1}$$

Transform

Detail solution

Let $u = 2 \cos{\left(x \right)} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 \cos{\left(x \right)} + 1\right)$ :
1. Differentiate $2 \cos{\left(x \right)} + 1$ 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 cosine is negative sine:
      
      $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
    So, the result is: $- 2 \sin{\left(x \right)}$
  2. The derivative of the constant $1$ is zero.
  The result is: $- 2 \sin{\left(x \right)}$
The result of the chain rule is:

$- \frac{\sin{\left(x \right)}}{\sqrt{2 \cos{\left(x \right)} + 1}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    -sin(x)     
----------------
  ______________
\/ 2*cos(x) + 1

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

Simplify

The second derivative [src]

 /     2               \ 
 |  sin (x)            | 
-|------------ + cos(x)| 
 \1 + 2*cos(x)         / 
-------------------------
       ______________    
     \/ 1 + 2*cos(x)

$$- \frac{\cos{\left(x \right)} + \frac{\sin^{2}{\left(x \right)}}{2 \cos{\left(x \right)} + 1}}{\sqrt{2 \cos{\left(x \right)} + 1}}$$

Simplify

The third derivative [src]

/                           2      \       
|      3*cos(x)        3*sin (x)   |       
|1 - ------------ - ---------------|*sin(x)
|    1 + 2*cos(x)                 2|       
\                   (1 + 2*cos(x)) /       
-------------------------------------------
                ______________             
              \/ 1 + 2*cos(x)

$$\frac{\left(1 - \frac{3 \cos{\left(x \right)}}{2 \cos{\left(x \right)} + 1} - \frac{3 \sin^{2}{\left(x \right)}}{\left(2 \cos{\left(x \right)} + 1\right)^{2}}\right) \sin{\left(x \right)}}{\sqrt{2 \cos{\left(x \right)} + 1}}$$

Simplify

The graph

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Derivative of (2cosx+1)^(1/2)

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The solution