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Derivative of:
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Identical expressions
y=ln(x^ two +5x+ six)
y equally ln(x squared plus 5x plus 6)
y equally ln(x to the power of two plus 5x plus six)
y=ln(x2+5x+6)
y=lnx2+5x+6
y=ln(x²+5x+6)
y=ln(x to the power of 2+5x+6)
y=lnx^2+5x+6
Similar expressions
y=ln(x^2+5x-6)
y=ln(x^2-5x+6)

The derivative of the function/ y=ln(x^2+5x+6)

Derivative of y=ln(x^2+5x+6)

The solution

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   / 2          \
log\x  + 5*x + 6/

$$\log{\left(\left(x^{2} + 5 x\right) + 6 \right)}$$

log(x^2 + 5*x + 6)

Detail solution

Let $u = \left(x^{2} + 5 x\right) + 6$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 5 x\right) + 6\right)$ :
1. Differentiate $\left(x^{2} + 5 x\right) + 6$ term by term:
  1. Differentiate $x^{2} + 5 x$ 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: $5$
    The result is: $2 x + 5$
  2. The derivative of the constant $6$ is zero.
  The result is: $2 x + 5$
The result of the chain rule is:

$\frac{2 x + 5}{\left(x^{2} + 5 x\right) + 6}$
Now simplify:

$\frac{2 x + 5}{x^{2} + 5 x + 6}$

The answer is:

$\frac{2 x + 5}{x^{2} + 5 x + 6}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  5 + 2*x   
------------
 2          
x  + 5*x + 6

$$\frac{2 x + 5}{\left(x^{2} + 5 x\right) + 6}$$

Simplify

The second derivative [src]

              2 
     (5 + 2*x)  
2 - ------------
         2      
    6 + x  + 5*x
----------------
       2        
  6 + x  + 5*x

$$\frac{- \frac{\left(2 x + 5\right)^{2}}{x^{2} + 5 x + 6} + 2}{x^{2} + 5 x + 6}$$

Simplify

The third derivative [src]

  /               2 \          
  |      (5 + 2*x)  |          
2*|-3 + ------------|*(5 + 2*x)
  |          2      |          
  \     6 + x  + 5*x/          
-------------------------------
                      2        
        /     2      \         
        \6 + x  + 5*x/

$$\frac{2 \left(2 x + 5\right) \left(\frac{\left(2 x + 5\right)^{2}}{x^{2} + 5 x + 6} - 3\right)}{\left(x^{2} + 5 x + 6\right)^{2}}$$

Simplify

The graph

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Derivative of y=ln(x^2+5x+6)

The graph:

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The solution