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Identical expressions
log4(x^ two +14x+ three hundred and five)+ nine
logarithm of 4(x squared plus 14x plus 305) plus 9
logarithm of 4(x to the power of two plus 14x plus three hundred and five) plus nine
log4(x2+14x+305)+9
log4x2+14x+305+9
log4(x²+14x+305)+9
log4(x to the power of 2+14x+305)+9
log4x^2+14x+305+9
Similar expressions
log4(x^2+14x-305)+9
log4(x^2-14x+305)+9
log4(x^2+14x+305)-9

The derivative of the function/ log4(x^2+14x+305)+9

Derivative of log4(x^2+14x+305)+9

The solution

You have entered [src]

   / 2             \    
log\x  + 14*x + 305/    
-------------------- + 9
       log(4)

$$\frac{\log{\left(\left(x^{2} + 14 x\right) + 305 \right)}}{\log{\left(4 \right)}} + 9$$

log(x^2 + 14*x + 305)/log(4) + 9

Detail solution

Differentiate $\frac{\log{\left(\left(x^{2} + 14 x\right) + 305 \right)}}{\log{\left(4 \right)}} + 9$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \left(x^{2} + 14 x\right) + 305$ .
  2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(\left(x^{2} + 14 x\right) + 305\right)$ :
    1. Differentiate $\left(x^{2} + 14 x\right) + 305$ term by term:
      1. Differentiate $x^{2} + 14 x$ term by term:
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        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: $14$
        
        The result is: $2 x + 14$
      2. The derivative of the constant $305$ is zero.
      The result is: $2 x + 14$
    The result of the chain rule is:
    
    $\frac{2 x + 14}{\left(x^{2} + 14 x\right) + 305}$
  So, the result is: $\frac{2 x + 14}{\left(\left(x^{2} + 14 x\right) + 305\right) \log{\left(4 \right)}}$
2. The derivative of the constant $9$ is zero.
The result is: $\frac{2 x + 14}{\left(\left(x^{2} + 14 x\right) + 305\right) \log{\left(4 \right)}}$
Now simplify:

$\frac{x + 7}{\left(x^{2} + 14 x + 305\right) \log{\left(2 \right)}}$

The answer is:

$\frac{x + 7}{\left(x^{2} + 14 x + 305\right) \log{\left(2 \right)}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

        14 + 2*x        
------------------------
/ 2             \       
\x  + 14*x + 305/*log(4)

$$\frac{2 x + 14}{\left(\left(x^{2} + 14 x\right) + 305\right) \log{\left(4 \right)}}$$

Simplify

The second derivative [src]

  /                2  \ 
  |       2*(7 + x)   | 
2*|1 - ---------------| 
  |           2       | 
  \    305 + x  + 14*x/ 
------------------------
/       2       \       
\305 + x  + 14*x/*log(4)

$$\frac{2 \left(- \frac{2 \left(x + 7\right)^{2}}{x^{2} + 14 x + 305} + 1\right)}{\left(x^{2} + 14 x + 305\right) \log{\left(4 \right)}}$$

Simplify

The third derivative [src]

  /                 2  \        
  |        4*(7 + x)   |        
4*|-3 + ---------------|*(7 + x)
  |            2       |        
  \     305 + x  + 14*x/        
--------------------------------
                    2           
   /       2       \            
   \305 + x  + 14*x/ *log(4)

$$\frac{4 \left(x + 7\right) \left(\frac{4 \left(x + 7\right)^{2}}{x^{2} + 14 x + 305} - 3\right)}{\left(x^{2} + 14 x + 305\right)^{2} \log{\left(4 \right)}}$$

Simplify

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Derivative of log4(x^2+14x+305)+9

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The solution