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Identical expressions
(one +lnx)/(x^ two +x+ one)
(1 plus lnx) divide by (x squared plus x plus 1)
(one plus lnx) divide by (x to the power of two plus x plus one)
(1+lnx)/(x2+x+1)
1+lnx/x2+x+1
(1+lnx)/(x²+x+1)
(1+lnx)/(x to the power of 2+x+1)
1+lnx/x^2+x+1
(1+lnx) divide by (x^2+x+1)
Similar expressions
(1+lnx)/(x^2+x-1)
(1-lnx)/(x^2+x+1)
(1+lnx)/(x^2-x+1)

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

Derivative of (1+lnx)/(x^2+x+1)

The solution

You have entered [src]

1 + log(x)
----------
 2        
x  + x + 1

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

(1 + log(x))/(x^2 + x + 1)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  3. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x + 1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

      1          (1 + log(x))*(-1 - 2*x)
-------------- + -----------------------
  / 2        \                    2     
x*\x  + x + 1/        / 2        \      
                      \x  + x + 1/

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

Simplify

The second derivative [src]

                                       /              2\
                                       |     (1 + 2*x) |
                        2*(1 + log(x))*|-1 + ----------|
                                       |              2|
  1     2*(1 + 2*x)                    \     1 + x + x /
- -- - -------------- + --------------------------------
   2     /         2\                       2           
  x    x*\1 + x + x /              1 + x + x            
--------------------------------------------------------
                                2                       
                       1 + x + x

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

Simplify

The third derivative [src]

                         /              2\                            /              2\
                         |     (1 + 2*x) |                            |     (1 + 2*x) |
                       6*|-1 + ----------|   6*(1 + 2*x)*(1 + log(x))*|-2 + ----------|
                         |              2|                            |              2|
2      3*(1 + 2*x)       \     1 + x + x /                            \     1 + x + x /
-- + --------------- + ------------------- - ------------------------------------------
 3    2 /         2\        /         2\                               2               
x    x *\1 + x + x /      x*\1 + x + x /                   /         2\                
                                                           \1 + x + x /                
---------------------------------------------------------------------------------------
                                                2                                      
                                       1 + x + x

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

Simplify

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Identical expressions

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Derivative of (1+lnx)/(x^2+x+1)

The graph:

Piecewise:

The solution