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The derivative of the function/ x×lnx-ln5×log5(x)

Derivative of x×lnx-ln5×log5(x)

The solution

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                  log(x)
x*log(x) - log(5)*------
                  log(5)

$$x \log{\left(x \right)} - \frac{\log{\left(x \right)}}{\log{\left(5 \right)}} \log{\left(5 \right)}$$

x*log(x) - log(5)*log(x)/log(5)

Detail solution

Differentiate $x \log{\left(x \right)} - \frac{\log{\left(x \right)}}{\log{\left(5 \right)}} \log{\left(5 \right)}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $\log{\left(x \right)} + 1$
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
    So, the result is: $\frac{1}{x \log{\left(5 \right)}}$
  So, the result is: $- \frac{1}{x}$
The result is: $\log{\left(x \right)} + 1 - \frac{1}{x}$

The answer is:

$\log{\left(x \right)} + 1 - \frac{1}{x}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1         
1 - - + log(x)
    x

$$\log{\left(x \right)} + 1 - \frac{1}{x}$$

Simplify

The second derivative [src]

    1
1 + -
    x
-----
  x

$$\frac{1 + \frac{1}{x}}{x}$$

Simplify

The third derivative [src]

 /    2\ 
-|1 + -| 
 \    x/ 
---------
     2   
    x

$$- \frac{1 + \frac{2}{x}}{x^{2}}$$

Simplify