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

Derivative of x^2/(√(1-x^2))

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
      2    
     x     
-----------
   ________
  /      2 
\/  1 - x
x21−x2\frac{x^{2}}{\sqrt{1 - x^{2}}} 
x^2/sqrt(1 - x^2)
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=x2f{\left(x \right)} = x^{2} and g(x)=1−x2g{\left(x \right)} = \sqrt{1 - x^{2}}.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: x2x^{2} goes to 2x2 x

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=1−x2u = 1 - x^{2}.

    2. Apply the power rule: u\sqrt{u} goes to 12u\frac{1}{2 \sqrt{u}}

    3. Then, apply the chain rule. Multiply by ddx(1−x2)\frac{d}{d x} \left(1 - x^{2}\right):

      1. Differentiate 1−x21 - x^{2} term by term:

        1. The derivative of the constant 11 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: x2x^{2} goes to 2x2 x

          So, the result is: −2x- 2 x

        The result is: −2x- 2 x

      The result of the chain rule is:

      −x1−x2- \frac{x}{\sqrt{1 - x^{2}}}

    Now plug in to the quotient rule:

    x31−x2+2x1−x21−x2\frac{\frac{x^{3}}{\sqrt{1 - x^{2}}} + 2 x \sqrt{1 - x^{2}}}{1 - x^{2}}

  2. Now simplify:

    x(2−x2)(1−x2)32\frac{x \left(2 - x^{2}\right)}{\left(1 - x^{2}\right)^{\frac{3}{2}}}

The answer is:

x(2−x2)(1−x2)32\frac{x \left(2 - x^{2}\right)}{\left(1 - x^{2}\right)^{\frac{3}{2}}}

The graph
02468-8-6-4-2-1010-5050
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
x3(1−x2)32+2x1−x2\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
−x2(3x2x2−1−1)1−x2+4x21−x2+21−x2\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 /
3x(−6x2x2−1−x2(5x2x2−1−3)1−x2+4)(1−x2)32\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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