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Identical expressions
(x^2sin2x)/(sqrt(x+ one))
(x squared sinus of 2x) divide by ( square root of (x plus 1))
(x squared sinus of 2x) divide by ( square root of (x plus one))
(x^2sin2x)/(√(x+1))
(x2sin2x)/(sqrt(x+1))
x2sin2x/sqrtx+1
(x²sin2x)/(sqrt(x+1))
(x to the power of 2sin2x)/(sqrt(x+1))
x^2sin2x/sqrtx+1
(x^2sin2x) divide by (sqrt(x+1))
Similar expressions
(x^2sin2x)/(sqrt(x-1))

The derivative of the function/ (x^2sin2x)/(sqrt(x+1))

Derivative of (x^2sin2x)/(sqrt(x+1))

The solution

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 2         
x *sin(2*x)
-----------
   _______ 
 \/ x + 1

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

(x^2*sin(2*x))/sqrt(x + 1)

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{2} \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \sqrt{x + 1}$ .

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

                                                                             2         
                2                           2*x*(x*cos(2*x) + sin(2*x))   3*x *sin(2*x)
2*sin(2*x) - 4*x *sin(2*x) + 8*x*cos(2*x) - --------------------------- + -------------
                                                       1 + x                         2 
                                                                            4*(1 + x)  
---------------------------------------------------------------------------------------
                                         _______                                       
                                       \/ 1 + x

$$\frac{- 4 x^{2} \sin{\left(2 x \right)} + \frac{3 x^{2} \sin{\left(2 x \right)}}{4 \left(x + 1\right)^{2}} + 8 x \cos{\left(2 x \right)} - \frac{2 x \left(x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right)}{x + 1} + 2 \sin{\left(2 x \right)}}{\sqrt{x + 1}}$$

Simplify

The third derivative [src]

                                                /     2                                   \       2                                       
                                 2            3*\- 2*x *sin(2*x) + 4*x*cos(2*x) + sin(2*x)/   15*x *sin(2*x)   9*x*(x*cos(2*x) + sin(2*x))
12*cos(2*x) - 24*x*sin(2*x) - 8*x *cos(2*x) - --------------------------------------------- - -------------- + ---------------------------
                                                                  1 + x                                  3                       2        
                                                                                                8*(1 + x)               2*(1 + x)         
------------------------------------------------------------------------------------------------------------------------------------------
                                                                  _______                                                                 
                                                                \/ 1 + x

$$\frac{- 8 x^{2} \cos{\left(2 x \right)} - \frac{15 x^{2} \sin{\left(2 x \right)}}{8 \left(x + 1\right)^{3}} - 24 x \sin{\left(2 x \right)} + \frac{9 x \left(x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right)}{2 \left(x + 1\right)^{2}} + 12 \cos{\left(2 x \right)} - \frac{3 \left(- 2 x^{2} \sin{\left(2 x \right)} + 4 x \cos{\left(2 x \right)} + \sin{\left(2 x \right)}\right)}{x + 1}}{\sqrt{x + 1}}$$

Simplify

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Derivative of (x^2sin2x)/(sqrt(x+1))

The graph:

Piecewise:

The solution