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The derivative of the function/ xsqrt(1-x^4)

Derivative of xsqrt(1-x^4)

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

You have entered [src] 
     ________
    /      4 
x*\/  1 - x
x1−x4x \sqrt{1 - x^{4}} 
x*sqrt(1 - x^4)
Detail solution

  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    g(x)=1−x4g{\left(x \right)} = \sqrt{1 - x^{4}}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=1−x4u = 1 - x^{4}.

    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−x4)\frac{d}{d x} \left(1 - x^{4}\right):

      1. Differentiate 1−x41 - x^{4} 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: x4x^{4} goes to 4x34 x^{3}

          So, the result is: −4x3- 4 x^{3}

        The result is: −4x3- 4 x^{3}

      The result of the chain rule is:

      −2x31−x4- \frac{2 x^{3}}{\sqrt{1 - x^{4}}}

    The result is: −2x41−x4+1−x4- \frac{2 x^{4}}{\sqrt{1 - x^{4}}} + \sqrt{1 - x^{4}}

  2. Now simplify:

    1−3x41−x4\frac{1 - 3 x^{4}}{\sqrt{1 - x^{4}}}

The answer is:

1−3x41−x4\frac{1 - 3 x^{4}}{\sqrt{1 - x^{4}}}

The graph
02468-8-6-4-2-10105-5
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   ________          4   
  /      4        2*x    
\/  1 - x   - -----------
                 ________
                /      4 
              \/  1 - x
−2x41−x4+1−x4- \frac{2 x^{4}}{\sqrt{1 - x^{4}}} + \sqrt{1 - x^{4}}
Simplify
The second derivative [src] 
     /          4 \
   3 |       2*x  |
2*x *|-5 + -------|
     |           4|
     \     -1 + x /
-------------------
       ________    
      /      4     
    \/  1 - x
2x3(2x4x4−1−5)1−x4\frac{2 x^{3} \left(\frac{2 x^{4}}{x^{4} - 1} - 5\right)}{\sqrt{1 - x^{4}}}
Simplify
The third derivative [src] 
     /           8           4 \
   2 |        4*x         8*x  |
6*x *|-5 - ---------- + -------|
     |              2         4|
     |     /      4\    -1 + x |
     \     \-1 + x /           /
--------------------------------
             ________           
            /      4            
          \/  1 - x
6x2(−4x8(x4−1)2+8x4x4−1−5)1−x4\frac{6 x^{2} \left(- \frac{4 x^{8}}{\left(x^{4} - 1\right)^{2}} + \frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\sqrt{1 - x^{4}}}
Simplify
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