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Identical expressions
sqrt((four hundred * nine . eighty-one)/((ten *sin(two x))+(two * one . eight *(cos(x))^2)))
square root of ((400 multiply by 9.81) divide by ((10 multiply by sinus of (2x)) plus (2 multiply by 1.8 multiply by ( co sinus of e of (x)) squared )))
square root of ((four hundred multiply by nine . eighty minus one) divide by ((ten multiply by sinus of (two x)) plus (two multiply by one . eight multiply by ( co sinus of e of (x)) squared )))
√((400*9.81)/((10*sin(2x))+(2*1.8*(cos(x))^2)))
sqrt((400*9.81)/((10*sin(2x))+(2*1.8*(cos(x))2)))
sqrt400*9.81/10*sin2x+2*1.8*cosx2
sqrt((400*9.81)/((10*sin(2x))+(2*1.8*(cos(x))²)))
sqrt((400*9.81)/((10*sin(2x))+(2*1.8*(cos(x)) to the power of 2)))
sqrt((4009.81)/((10sin(2x))+(21.8(cos(x))^2)))
sqrt((4009.81)/((10sin(2x))+(21.8(cos(x))2)))
sqrt4009.81/10sin2x+21.8cosx2
sqrt4009.81/10sin2x+21.8cosx^2
sqrt((400*9.81) divide by ((10*sin(2x))+(2*1.8*(cos(x))^2)))
Similar expressions
sqrt((400*9.81)/((10*sin(2x))-(2*1.8*(cos(x))^2)))
sqrt((400*9.81)/((10*sin(2x))+(2*1.8*(cosx)^2)))

The derivative of the function/ sqrt((400*9.81)/((10*sin(2x))+(2*1.8*(cos(x))^2)))

Derivative of sqrt((4009.81)/((10sin(2x))+(21.8(cos(x))^2)))

The solution

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        ___________________________
       /         /981*400\         
      /          |-------|         
     /           \  100  /         
    /    ------------------------- 
   /                   9*2    2    
  /      10*sin(2*x) + ---*cos (x) 
\/                      5

$$\sqrt{\frac{\frac{981}{100} \cdot 400}{10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}}}$$

sqrt((981*400/100)/(10*sin(2*x) + (9*2/5)*cos(x)^2))

Detail solution

Let $u = \frac{\frac{981}{100} \cdot 400}{10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}}$ .
Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{\frac{981}{100} \cdot 400}{10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}}$ :
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = 10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}\right)$ :
    1. Differentiate $10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}$ term by term:
      1. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = 2 x$ .
        
        The derivative of sine is cosine:
        
        $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
        
        The derivative of a constant times a function is the constant times the derivative of the function.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $2$
        
        The result of the chain rule is:
        
        $2 \cos{\left(2 x \right)}$
        
        So, the result is: $20 \cos{\left(2 x \right)}$
      2. The derivative of a constant times a function is the constant times the derivative of the function.
        
        Let $u = \cos{\left(x \right)}$ .
        
        Apply the power rule: $u^{2}$ goes to $2 u$
        
        Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
        
        The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
        
        The result of the chain rule is:
        
        $- 2 \sin{\left(x \right)} \cos{\left(x \right)}$
        
        So, the result is: $- \frac{36 \sin{\left(x \right)} \cos{\left(x \right)}}{5}$
      The result is: $- \frac{36 \sin{\left(x \right)} \cos{\left(x \right)}}{5} + 20 \cos{\left(2 x \right)}$
    The result of the chain rule is:
    
    $- \frac{- \frac{36 \sin{\left(x \right)} \cos{\left(x \right)}}{5} + 20 \cos{\left(2 x \right)}}{\left(10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}\right)^{2}}$
  So, the result is: $- \frac{3924 \left(- \frac{36 \sin{\left(x \right)} \cos{\left(x \right)}}{5} + 20 \cos{\left(2 x \right)}\right)}{\left(10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}\right)^{2}}$
The result of the chain rule is:

$- \frac{3 \sqrt{109} \left(- \frac{36 \sin{\left(x \right)} \cos{\left(x \right)}}{5} + 20 \cos{\left(2 x \right)}\right)}{\left(10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}\right)^{2} \sqrt{\frac{1}{10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}}}}$
Now simplify:

$\frac{\sqrt{1090} \left(\frac{27 \sin{\left(2 x \right)}}{2} - 75 \cos{\left(2 x \right)}\right)}{\left(25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}\right)^{2} \sqrt{\frac{1}{25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}}}}$

The answer is:

$\frac{\sqrt{1090} \left(\frac{27 \sin{\left(2 x \right)}}{2} - 75 \cos{\left(2 x \right)}\right)}{\left(25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}\right)^{2} \sqrt{\frac{1}{25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

           ___________________________                                   /   2                \
          /            3924            /               36*cos(x)*sin(x)\ |cos (x)   5*sin(2*x)|
1962*    /  ------------------------- *|-20*cos(2*x) + ----------------|*|------- + ----------|
        /                 9*2    2     \                      5        / \  1090       1962   /
       /    10*sin(2*x) + ---*cos (x)                                                          
     \/                    5                                                                   
-----------------------------------------------------------------------------------------------
                                                             2                                 
                                  /              9*2    2   \                                  
                                  |10*sin(2*x) + ---*cos (x)|                                  
                                  \               5         /

$$\frac{1962 \sqrt{\frac{3924}{10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}}} \left(\frac{36 \sin{\left(x \right)} \cos{\left(x \right)}}{5} - 20 \cos{\left(2 x \right)}\right) \left(\frac{5 \sin{\left(2 x \right)}}{1962} + \frac{\cos^{2}{\left(x \right)}}{1090}\right)}{\left(10 \sin{\left(2 x \right)} + \frac{2 \cdot 9}{5} \cos^{2}{\left(x \right)}\right)^{2}}$$

Simplify

The second derivative [src]

                 ________________________                                                                                                          
    _____       /           1             /                                  2   /     2                 \ /       2           2                 \\
3*\/ 218 *     /  ---------------------- *\3*(-25*cos(2*x) + 9*cos(x)*sin(x))  + \9*cos (x) + 25*sin(2*x)/*\- 9*sin (x) + 9*cos (x) + 50*sin(2*x)//
              /                     2                                                                                                              
             /                 9*cos (x)                                                                                                           
            /     5*sin(2*x) + ---------                                                                                                           
          \/                       5                                                                                                               
---------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                      2                                                            
                                                             /     2                 \                                                             
                                                             \9*cos (x) + 25*sin(2*x)/

$$\frac{3 \sqrt{218} \left(3 \left(9 \sin{\left(x \right)} \cos{\left(x \right)} - 25 \cos{\left(2 x \right)}\right)^{2} + \left(25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}\right) \left(- 9 \sin^{2}{\left(x \right)} + 50 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}\right)\right) \sqrt{\frac{1}{5 \sin{\left(2 x \right)} + \frac{9 \cos^{2}{\left(x \right)}}{5}}}}{\left(25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}\right)^{2}}$$

Simplify

The third derivative [src]

                 ________________________                                  /                                                                              2\
    _____       /           1                                              |        2            2                     15*(-25*cos(2*x) + 9*cos(x)*sin(x)) |
3*\/ 218 *     /  ---------------------- *(-25*cos(2*x) + 9*cos(x)*sin(x))*|- 81*sin (x) + 45*cos (x) + 350*sin(2*x) + ------------------------------------|
              /                     2                                      |                                                      2                        |
             /                 9*cos (x)                                   \                                                 9*cos (x) + 25*sin(2*x)       /
            /     5*sin(2*x) + ---------                                                                                                                    
          \/                       5                                                                                                                        
------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                          2                                                                 
                                                                 /     2                 \                                                                  
                                                                 \9*cos (x) + 25*sin(2*x)/

$$\frac{3 \sqrt{218} \left(9 \sin{\left(x \right)} \cos{\left(x \right)} - 25 \cos{\left(2 x \right)}\right) \left(\frac{15 \left(9 \sin{\left(x \right)} \cos{\left(x \right)} - 25 \cos{\left(2 x \right)}\right)^{2}}{25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}} - 81 \sin^{2}{\left(x \right)} + 350 \sin{\left(2 x \right)} + 45 \cos^{2}{\left(x \right)}\right) \sqrt{\frac{1}{5 \sin{\left(2 x \right)} + \frac{9 \cos^{2}{\left(x \right)}}{5}}}}{\left(25 \sin{\left(2 x \right)} + 9 \cos^{2}{\left(x \right)}\right)^{2}}$$

Simplify

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Derivative of sqrt((4009.81)/((10sin(2x))+(21.8(cos(x))^2)))

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt((400*9.81)/((10*sin(2x))+(2*1.8*(cos(x))^2)))

The graph:

Piecewise:

The solution

Derivative of sqrt((4009.81)/((10sin(2x))+(21.8(cos(x))^2)))