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The derivative of the function/ ln(-x^2-9x-6)

Derivative of ln(-x^2-9x-6)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
   /   2          \
log\- x  - 9*x - 6/
$$\log{\left(\left(- x^{2} - 9 x\right) - 6 \right)}$$ 
log(-x^2 - 9*x - 6)
Detail solution

  1. Let .

  2. The derivative of is .

  3. Then, apply the chain rule. Multiply by :

    1. Differentiate term by term:

      1. Differentiate term by term:

        1. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        2. The derivative of a constant times a function is the constant times the derivative of the function.

          1. Apply the power rule: goes to

          So, the result is:

        The result is:

      2. The derivative of the constant is zero.

      The result is:

    The result of the chain rule is:

  4. Now simplify:

The answer is:

The graph
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   -9 - 2*x   
--------------
   2          
- x  - 9*x - 6
$$\frac{- 2 x - 9}{\left(- x^{2} - 9 x\right) - 6}$$
Simplify
The second derivative [src] 
              2 
     (9 + 2*x)  
2 - ------------
         2      
    6 + x  + 9*x
----------------
       2        
  6 + x  + 9*x
$$\frac{- \frac{\left(2 x + 9\right)^{2}}{x^{2} + 9 x + 6} + 2}{x^{2} + 9 x + 6}$$
Simplify
The third derivative [src] 
  /               2 \          
  |      (9 + 2*x)  |          
2*|-3 + ------------|*(9 + 2*x)
  |          2      |          
  \     6 + x  + 9*x/          
-------------------------------
                      2        
        /     2      \         
        \6 + x  + 9*x/
$$\frac{2 \left(2 x + 9\right) \left(\frac{\left(2 x + 9\right)^{2}}{x^{2} + 9 x + 6} - 3\right)}{\left(x^{2} + 9 x + 6\right)^{2}}$$
Simplify
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