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Graphing y =:
ln(x^2+2x+2)
Identical expressions
ln(x^ two + two x+2)
ln(x squared plus 2x plus 2)
ln(x to the power of two plus two x plus 2)
ln(x2+2x+2)
lnx2+2x+2
ln(x²+2x+2)
ln(x to the power of 2+2x+2)
lnx^2+2x+2
Similar expressions
ln(x^2+2x-2)
ln(x^2-2x+2)

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

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

The solution

You have entered [src]

   / 2          \
log\x  + 2*x + 2/

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

log(x^2 + 2*x + 2)

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  2 + 2*x   
------------
 2          
x  + 2*x + 2

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

          /               2 \
          |      4*(1 + x)  |
4*(1 + x)*|-3 + ------------|
          |          2      |
          \     2 + x  + 2*x/
-----------------------------
                     2       
       /     2      \        
       \2 + x  + 2*x/

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

Simplify

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Identical expressions

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

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Piecewise:

The solution