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y=ln^2((x^2)+1)
The derivative of the function/ y=ln^2((x^2)+1)

Derivative of y=ln^2((x^2)+1)

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

You have entered [src] 
   2/ 2    \
log \x  + 1/
log⁡(x2+1)2\log{\left(x^{2} + 1 \right)}^{2} 
log(x^2 + 1)^2
Detail solution

  1. Let u=log⁡(x2+1)u = \log{\left(x^{2} + 1 \right)}.

  2. Apply the power rule: u2u^{2} goes to 2u2 u

  3. Then, apply the chain rule. Multiply by ddxlog⁡(x2+1)\frac{d}{d x} \log{\left(x^{2} + 1 \right)}:

    1. Let u=x2+1u = x^{2} + 1.

    2. The derivative of log⁡(u)\log{\left(u \right)} is 1u\frac{1}{u}.

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

      1. Differentiate x2+1x^{2} + 1 term by term:

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

        2. The derivative of the constant 11 is zero.

        The result is: 2x2 x

      The result of the chain rule is:

      2xx2+1\frac{2 x}{x^{2} + 1}

    The result of the chain rule is:

    4xlog⁡(x2+1)x2+1\frac{4 x \log{\left(x^{2} + 1 \right)}}{x^{2} + 1}

  4. Now simplify:

    4xlog⁡(x2+1)x2+1\frac{4 x \log{\left(x^{2} + 1 \right)}}{x^{2} + 1}

The answer is:

4xlog⁡(x2+1)x2+1\frac{4 x \log{\left(x^{2} + 1 \right)}}{x^{2} + 1}

The graph
02468-8-6-4-2-1010-2525
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
       / 2    \
4*x*log\x  + 1/
---------------
      2        
     x  + 1
4xlog⁡(x2+1)x2+1\frac{4 x \log{\left(x^{2} + 1 \right)}}{x^{2} + 1}
Simplify
The second derivative [src] 
  /    2       2    /     2\              \
  | 2*x     2*x *log\1 + x /      /     2\|
4*|------ - ---------------- + log\1 + x /|
  |     2             2                   |
  \1 + x         1 + x                    /
-------------------------------------------
                        2                  
                   1 + x
4(−2x2log⁡(x2+1)x2+1+2x2x2+1+log⁡(x2+1))x2+1\frac{4 \left(- \frac{2 x^{2} \log{\left(x^{2} + 1 \right)}}{x^{2} + 1} + \frac{2 x^{2}}{x^{2} + 1} + \log{\left(x^{2} + 1 \right)}\right)}{x^{2} + 1}
Simplify
The third derivative [src] 
    /                        2       2    /     2\\
    |         /     2\    6*x     4*x *log\1 + x /|
8*x*|3 - 3*log\1 + x / - ------ + ----------------|
    |                         2             2     |
    \                    1 + x         1 + x      /
---------------------------------------------------
                             2                     
                     /     2\                      
                     \1 + x /
8x(4x2log⁡(x2+1)x2+1−6x2x2+1−3log⁡(x2+1)+3)(x2+1)2\frac{8 x \left(\frac{4 x^{2} \log{\left(x^{2} + 1 \right)}}{x^{2} + 1} - \frac{6 x^{2}}{x^{2} + 1} - 3 \log{\left(x^{2} + 1 \right)} + 3\right)}{\left(x^{2} + 1\right)^{2}}
Simplify
The graph
Derivative of y=ln^2((x^2)+1)
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