Derivative of y=ln(x²+3x+2)

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   / 2          \
log\x  + 3*x + 2/

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

log(x^2 + 3*x + 2)

Detail solution

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

$\frac{2 x + 3}{\left(x^{2} + 3 x\right) + 2}$
Now simplify:

$\frac{2 x + 3}{x^{2} + 3 x + 2}$

The answer is:

$\frac{2 x + 3}{x^{2} + 3 x + 2}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  3 + 2*x   
------------
 2          
x  + 3*x + 2

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

Simplify

The second derivative [src]

              2 
     (3 + 2*x)  
2 - ------------
         2      
    2 + x  + 3*x
----------------
       2        
  2 + x  + 3*x

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

Simplify

3-я производная [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=ln(x²+3x+2)

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The solution