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Identical expressions
log6(x^ two + two 4x+ one hundred and forty-seven)+2
logarithm of 6(x squared plus 24x plus 147) plus 2
logarithm of 6(x to the power of two plus two 4x plus one hundred and forty minus seven) plus 2
log6(x2+24x+147)+2
log6x2+24x+147+2
log6(x²+24x+147)+2
log6(x to the power of 2+24x+147)+2
log6x^2+24x+147+2
Similar expressions
log6(x^2+24x-147)+2
log6(x^2-24x+147)+2
log6(x^2+24x+147)-2

The derivative of the function/ log6(x^2+24x+147)+2

Derivative of log6(x^2+24x+147)+2

The solution

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   / 2             \    
log\x  + 24*x + 147/    
-------------------- + 2
       log(6)

$$\frac{\log{\left(x^{2} + 24 x + 147 \right)}}{\log{\left(6 \right)}} + 2$$

  /   / 2             \    \
d |log\x  + 24*x + 147/    |
--|-------------------- + 2|
dx\       log(6)           /

$$\frac{d}{d x} \left(\frac{\log{\left(x^{2} + 24 x + 147 \right)}}{\log{\left(6 \right)}} + 2\right)$$

Transform

Detail solution

Differentiate $\frac{\log{\left(x^{2} + 24 x + 147 \right)}}{\log{\left(6 \right)}} + 2$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = x^{2} + 24 x + 147$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} + 24 x + 147\right)$ :
    1. Differentiate $x^{2} + 24 x + 147$ 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.
        
        Apply the power rule: $x$ goes to $1$
        
        So, the result is: $24$
      3. The derivative of the constant $147$ is zero.
      The result is: $2 x + 24$
    The result of the chain rule is:
    
    $\frac{2 x + 24}{x^{2} + 24 x + 147}$
  So, the result is: $\frac{2 x + 24}{\left(x^{2} + 24 x + 147\right) \log{\left(6 \right)}}$
2. The derivative of the constant $2$ is zero.
The result is: $\frac{2 x + 24}{\left(x^{2} + 24 x + 147\right) \log{\left(6 \right)}}$
Now simplify:

$\frac{2 \left(x + 12\right)}{\left(x^{2} + 24 x + 147\right) \log{\left(6 \right)}}$

The answer is:

$\frac{2 \left(x + 12\right)}{\left(x^{2} + 24 x + 147\right) \log{\left(6 \right)}}$

The graph

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The first derivative [src]

        24 + 2*x        
------------------------
/ 2             \       
\x  + 24*x + 147/*log(6)

$$\frac{2 x + 24}{\left(x^{2} + 24 x + 147\right) \log{\left(6 \right)}}$$

Simplify

The second derivative [src]

  /                2  \ 
  |      2*(12 + x)   | 
2*|1 - ---------------| 
  |           2       | 
  \    147 + x  + 24*x/ 
------------------------
/       2       \       
\147 + x  + 24*x/*log(6)

$$\frac{2 \left(- \frac{2 \left(x + 12\right)^{2}}{x^{2} + 24 x + 147} + 1\right)}{\left(x^{2} + 24 x + 147\right) \log{\left(6 \right)}}$$

Simplify

The third derivative [src]

  /                 2  \         
  |       4*(12 + x)   |         
4*|-3 + ---------------|*(12 + x)
  |            2       |         
  \     147 + x  + 24*x/         
---------------------------------
                     2           
    /       2       \            
    \147 + x  + 24*x/ *log(6)

$$\frac{4 \left(x + 12\right) \left(\frac{4 \left(x + 12\right)^{2}}{x^{2} + 24 x + 147} - 3\right)}{\left(x^{2} + 24 x + 147\right)^{2} \log{\left(6 \right)}}$$

Simplify

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Derivative of log6(x^2+24x+147)+2

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The solution