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Derivative of ln^3(x^2+1)
Identical expressions
ln^ three (x^ two + one)
ln cubed (x squared plus 1)
ln to the power of three (x to the power of two plus one)
ln3(x2+1)
ln3x2+1
ln³(x²+1)
ln to the power of 3(x to the power of 2+1)
ln^3x^2+1
Similar expressions
ln^3(x^2-1)

The derivative of the function/ ln^3(x^2+1)

Derivative of ln^3(x^2+1)

The solution

You have entered [src]

   3/ 2    \
log \x  + 1/

$$\log{\left(x^{2} + 1 \right)}^{3}$$

log(x^2 + 1)^3

Detail solution

Let $u = \log{\left(x^{2} + 1 \right)}$ .
Apply the power rule: $u^{3}$ goes to $3 u^{2}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x^{2} + 1 \right)}$ :
1. Let $u = x^{2} + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 1\right)$ :
  1. Differentiate $x^{2} + 1$ term by term:
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    2. The derivative of the constant $1$ is zero.
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{2 x}{x^{2} + 1}$
The result of the chain rule is:

$\frac{6 x \log{\left(x^{2} + 1 \right)}^{2}}{x^{2} + 1}$
Now simplify:

$\frac{6 x \log{\left(x^{2} + 1 \right)}^{2}}{x^{2} + 1}$

The answer is:

$\frac{6 x \log{\left(x^{2} + 1 \right)}^{2}}{x^{2} + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2/ 2    \
6*x*log \x  + 1/
----------------
      2         
     x  + 1

$$\frac{6 x \log{\left(x^{2} + 1 \right)}^{2}}{x^{2} + 1}$$

Simplify

The second derivative [src]

  /    2       2    /     2\              \            
  | 4*x     2*x *log\1 + x /      /     2\|    /     2\
6*|------ - ---------------- + log\1 + x /|*log\1 + x /
  |     2             2                   |            
  \1 + x         1 + x                    /            
-------------------------------------------------------
                              2                        
                         1 + x

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

Simplify

The third derivative [src]

     /                                       2        2    /     2\      2    2/     2\\
     |       2/     2\        /     2\    4*x     12*x *log\1 + x /   4*x *log \1 + x /|
12*x*|- 3*log \1 + x / + 6*log\1 + x / + ------ - ----------------- + -----------------|
     |                                        2              2                   2     |
     \                                   1 + x          1 + x               1 + x      /
----------------------------------------------------------------------------------------
                                               2                                        
                                       /     2\                                         
                                       \1 + x /

$$\frac{12 x \left(\frac{4 x^{2} \log{\left(x^{2} + 1 \right)}^{2}}{x^{2} + 1} - \frac{12 x^{2} \log{\left(x^{2} + 1 \right)}}{x^{2} + 1} + \frac{4 x^{2}}{x^{2} + 1} - 3 \log{\left(x^{2} + 1 \right)}^{2} + 6 \log{\left(x^{2} + 1 \right)}\right)}{\left(x^{2} + 1\right)^{2}}$$

Simplify

The graph

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Derivative of ln^3(x^2+1)

The graph:

Piecewise:

The solution