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Derivative of:
Derivative of y=2x+1
Derivative of asin(t)
Derivative of 7/x^4
Derivative of x*sin(4*x)
Identical expressions
sqrt(one -sin^ two (x)* zero . one)
square root of (1 minus sinus of squared (x) multiply by 0.1)
square root of (one minus sinus of to the power of two (x) multiply by zero . one)
√(1-sin^2(x)*0.1)
sqrt(1-sin2(x)*0.1)
sqrt1-sin2x*0.1
sqrt(1-sin²(x)*0.1)
sqrt(1-sin to the power of 2(x)*0.1)
sqrt(1-sin^2(x)0.1)
sqrt(1-sin2(x)0.1)
sqrt1-sin2x0.1
sqrt1-sin^2x0.1
Similar expressions
sqrt(1+sin^2(x)*0.1)

The derivative of the function/ sqrt(1-sin^2(x)*0.1)

Derivative of sqrt(1-sin^2(x)*0.1)

The solution

You have entered [src]

     _____________
    /        2    
   /      sin (x) 
  /   1 - ------- 
\/           10

$$\sqrt{- \frac{\sin^{2}{\left(x \right)}}{10} + 1}$$

sqrt(1 - sin(x)^2/10)

Detail solution

Let $u = - \frac{\sin^{2}{\left(x \right)}}{10} + 1$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \frac{\sin^{2}{\left(x \right)}}{10} + 1\right)$ :
1. Differentiate $- \frac{\sin^{2}{\left(x \right)}}{10} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \sin{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      The result of the chain rule is:
      
      $2 \sin{\left(x \right)} \cos{\left(x \right)}$
    So, the result is: $- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{5}$
  The result is: $- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{5}$
The result of the chain rule is:

$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{10 \sqrt{- \frac{\sin^{2}{\left(x \right)}}{10} + 1}}$
Now simplify:

$- \frac{\sqrt{5} \sin{\left(2 x \right)}}{10 \sqrt{\cos{\left(2 x \right)} + 19}}$

The answer is:

$- \frac{\sqrt{5} \sin{\left(2 x \right)}}{10 \sqrt{\cos{\left(2 x \right)} + 19}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   -cos(x)*sin(x)    
---------------------
        _____________
       /        2    
      /      sin (x) 
10*  /   1 - ------- 
   \/           10

$$- \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{10 \sqrt{- \frac{\sin^{2}{\left(x \right)}}{10} + 1}}$$

Simplify

The second derivative [src]

                               2       2   
        2            2      cos (x)*sin (x)
- 10*cos (x) + 10*sin (x) - ---------------
                                     2     
                                  sin (x)  
                              1 - -------  
                                     10    
-------------------------------------------
                    _____________          
                   /        2              
                  /      sin (x)           
           100*  /   1 - -------           
               \/           10

$$\frac{10 \sin^{2}{\left(x \right)} - 10 \cos^{2}{\left(x \right)} - \frac{\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{1 - \frac{\sin^{2}{\left(x \right)}}{10}}}{100 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{10}}}$$

Simplify

The third derivative [src]

/             2             2           2       2   \              
|       30*cos (x)    30*sin (x)   3*cos (x)*sin (x)|              
|400 - ----------- + ----------- - -----------------|*cos(x)*sin(x)
|             2             2                     2 |              
|          sin (x)       sin (x)     /       2   \  |              
|      1 - -------   1 - -------     |    sin (x)|  |              
|             10            10       |1 - -------|  |              
\                                    \       10  /  /              
-------------------------------------------------------------------
                                _____________                      
                               /        2                          
                              /      sin (x)                       
                      1000*  /   1 - -------                       
                           \/           10

$$\frac{\left(400 + \frac{30 \sin^{2}{\left(x \right)}}{1 - \frac{\sin^{2}{\left(x \right)}}{10}} - \frac{30 \cos^{2}{\left(x \right)}}{1 - \frac{\sin^{2}{\left(x \right)}}{10}} - \frac{3 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{10}\right)^{2}}\right) \sin{\left(x \right)} \cos{\left(x \right)}}{1000 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{10}}}$$

Simplify

4-я производная [src]

                                     4             4            4       2           4       4            2       4             2       2   
         2             2       60*cos (x)    60*sin (x)   36*cos (x)*sin (x)   3*cos (x)*sin (x)   36*cos (x)*sin (x)   440*cos (x)*sin (x)
- 800*sin (x) + 800*cos (x) - ----------- - ----------- - ------------------ - ----------------- + ------------------ + -------------------
                                     2             2                     2                    3                   2                2       
                                  sin (x)       sin (x)     /       2   \        /       2   \       /       2   \              sin (x)    
                              1 - -------   1 - -------     |    sin (x)|        |    sin (x)|       |    sin (x)|          1 - -------    
                                     10            10       |1 - -------|        |1 - -------|       |1 - -------|                 10      
                                                            \       10  /        \       10  /       \       10  /                         
-------------------------------------------------------------------------------------------------------------------------------------------
                                                                    _____________                                                          
                                                                   /        2                                                              
                                                                  /      sin (x)                                                           
                                                          2000*  /   1 - -------                                                           
                                                               \/           10

$$\frac{- 800 \sin^{2}{\left(x \right)} + 800 \cos^{2}{\left(x \right)} - \frac{60 \sin^{4}{\left(x \right)}}{1 - \frac{\sin^{2}{\left(x \right)}}{10}} + \frac{440 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{1 - \frac{\sin^{2}{\left(x \right)}}{10}} - \frac{60 \cos^{4}{\left(x \right)}}{1 - \frac{\sin^{2}{\left(x \right)}}{10}} + \frac{36 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{10}\right)^{2}} - \frac{36 \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{10}\right)^{2}} - \frac{3 \sin^{4}{\left(x \right)} \cos^{4}{\left(x \right)}}{\left(1 - \frac{\sin^{2}{\left(x \right)}}{10}\right)^{3}}}{2000 \sqrt{1 - \frac{\sin^{2}{\left(x \right)}}{10}}}$$

Simplify

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Derivative of sqrt(1-sin^2(x)*0.1)

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The solution