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x^2/(√(1-x^2))
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Similar expressions
x^2/(√(1+x^2))

The derivative of the function/ x^2/(√(1-x^2))

Derivative of x^2/(√(1-x^2))

The solution

You have entered [src]

      2    
     x     
-----------
   ________
  /      2 
\/  1 - x

$$\frac{x^{2}}{\sqrt{1 - x^{2}}}$$

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

Detail solution

Apply the quotient rule, which is:

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

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

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

$\frac{\frac{x^{3}}{\sqrt{1 - x^{2}}} + 2 x \sqrt{1 - x^{2}}}{1 - x^{2}}$
Now simplify:

$\frac{x \left(2 - x^{2}\right)}{\left(1 - x^{2}\right)^{\frac{3}{2}}}$

The answer is:

$\frac{x \left(2 - x^{2}\right)}{\left(1 - x^{2}\right)^{\frac{3}{2}}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      3                  
     x            2*x    
----------- + -----------
        3/2      ________
/     2\        /      2 
\1 - x /      \/  1 - x

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

Simplify

The second derivative [src]

                /          2 \
              2 |       3*x  |
             x *|-1 + -------|
        2       |           2|
     4*x        \     -1 + x /
2 + ------ - -----------------
         2              2     
    1 - x          1 - x      
------------------------------
            ________          
           /      2           
         \/  1 - x

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

Simplify

The third derivative [src]

    /                 /          2 \\
    |               2 |       5*x  ||
    |              x *|-3 + -------||
    |         2       |           2||
    |      6*x        \     -1 + x /|
3*x*|4 - ------- - -----------------|
    |          2              2     |
    \    -1 + x          1 - x      /
-------------------------------------
                     3/2             
             /     2\                
             \1 - x /

$$\frac{3 x \left(- \frac{6 x^{2}}{x^{2} - 1} - \frac{x^{2} \left(\frac{5 x^{2}}{x^{2} - 1} - 3\right)}{1 - x^{2}} + 4\right)}{\left(1 - x^{2}\right)^{\frac{3}{2}}}$$

Simplify

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Derivative of x^2/(√(1-x^2))

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The solution