Derivative of 3log2(x)(4x+5)-6x

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  log(x)                
3*------*(4*x + 5) - 6*x
  log(2)

- 6 x + 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \left(4 x + 5\right)

(3*(log(x)/log(2)))*(4*x + 5) - 6*x

Detail solution

Differentiate $- 6 x + 3 \frac{\log{\left(x \right)}}{\log{\left(2 \right)}} \left(4 x + 5\right)$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 3 \left(4 x + 5\right) \log{\left(x \right)}$ and $g{\left(x \right)} = \log{\left(2 \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = 4 x + 5$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Differentiate $4 x + 5$ term by term:
        
        The derivative of the constant $5$ is zero.
        
        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: $4$
        
        The result is: $4$
      $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result is: $4 \log{\left(x \right)} + \frac{4 x + 5}{x}$
    So, the result is: $12 \log{\left(x \right)} + \frac{3 \left(4 x + 5\right)}{x}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of the constant $\log{\left(2 \right)}$ is zero.
  Now plug in to the quotient rule:
  
  $\frac{12 \log{\left(x \right)} + \frac{3 \left(4 x + 5\right)}{x}}{\log{\left(2 \right)}}$
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: $-6$
The result is: $\frac{12 \log{\left(x \right)} + \frac{3 \left(4 x + 5\right)}{x}}{\log{\left(2 \right)}} - 6$
Now simplify:

$\frac{12 \log{\left(x \right)}}{\log{\left(2 \right)}} - 6 + \frac{12}{\log{\left(2 \right)}} + \frac{15}{x \log{\left(2 \right)}}$

The answer is:

\frac{12 \log{\left(x \right)}}{\log{\left(2 \right)}} - 6 + \frac{12}{\log{\left(2 \right)}} + \frac{15}{x \log{\left(2 \right)}}

The graph

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Plot the graph f'(x)

The first derivative [src]

     12*log(x)   3*(4*x + 5)
-6 + --------- + -----------
       log(2)      x*log(2)

\frac{12 \log{\left(x \right)}}{\log{\left(2 \right)}} - 6 + \frac{3 \left(4 x + 5\right)}{x \log{\left(2 \right)}}

Simplify

The second derivative [src]

  /    5 + 4*x\
3*|8 - -------|
  \       x   /
---------------
    x*log(2)

\frac{3 \left(8 - \frac{4 x + 5}{x}\right)}{x \log{\left(2 \right)}}

Simplify

The third derivative [src]

  /     5 + 4*x\
6*|-6 + -------|
  \        x   /
----------------
    2           
   x *log(2)

\frac{6 \left(-6 + \frac{4 x + 5}{x}\right)}{x^{2} \log{\left(2 \right)}}

Simplify

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Derivative of 3log2(x)(4x+5)-6x

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