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Identical expressions
y=(sin(2x)+ one)^ zero , five
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Similar expressions
y=(sin(2x)-1)^0,5

The derivative of the function/ y=(sin(2x)+1)^0,5

Derivative of y=(sin(2x)+1)^0,5

The solution

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  ______________
\/ sin(2*x) + 1

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

sqrt(sin(2*x) + 1)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

/            2                      \         
|       3*cos (2*x)      6*sin(2*x) |         
|-4 + --------------- + ------------|*cos(2*x)
|                   2   1 + sin(2*x)|         
\     (1 + sin(2*x))                /         
----------------------------------------------
                 ______________               
               \/ 1 + sin(2*x)

$$\frac{\left(-4 + \frac{6 \sin{\left(2 x \right)}}{\sin{\left(2 x \right)} + 1} + \frac{3 \cos^{2}{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} + 1\right)^{2}}\right) \cos{\left(2 x \right)}}{\sqrt{\sin{\left(2 x \right)} + 1}}$$

Simplify

The graph

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Derivative of y=(sin(2x)+1)^0,5

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The solution