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Integral of d{x}:
sqrt(1+sin^2(x))
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Similar expressions
sqrt(1-sin^2(x))

The derivative of the function/ sqrt(1+sin^2(x))

Derivative of sqrt(1+sin^2(x))

The solution

You have entered [src]

   _____________
  /        2    
\/  1 + sin (x)

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

sqrt(1 + sin(x)^2)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of sqrt(1+sin^2(x))

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The solution