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The derivative of the function/ 1/x+2ln(x)-(ln(x)/x)

Derivative of 1/x+2ln(x)-(ln(x)/x)

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
1              log(x)
- + 2*log(x) - ------
x                x
(2log⁡(x)+1x)−log⁡(x)x\left(2 \log{\left(x \right)} + \frac{1}{x}\right) - \frac{\log{\left(x \right)}}{x} 
1/x + 2*log(x) - log(x)/x
Detail solution

  1. Differentiate (2log⁡(x)+1x)−log⁡(x)x\left(2 \log{\left(x \right)} + \frac{1}{x}\right) - \frac{\log{\left(x \right)}}{x} term by term:

    1. Differentiate 2log⁡(x)+1x2 \log{\left(x \right)} + \frac{1}{x} term by term:

      1. Apply the power rule: 1x\frac{1}{x} goes to −1x2- \frac{1}{x^{2}}

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

        1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

        So, the result is: 2x\frac{2}{x}

      The result is: 2x−1x2\frac{2}{x} - \frac{1}{x^{2}}

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

      1. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)} and g(x)=xg{\left(x \right)} = x.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. Apply the power rule: xx goes to 11

        Now plug in to the quotient rule:

        1−log⁡(x)x2\frac{1 - \log{\left(x \right)}}{x^{2}}

      So, the result is: −1−log⁡(x)x2- \frac{1 - \log{\left(x \right)}}{x^{2}}

    The result is: 2x−1−log⁡(x)x2−1x2\frac{2}{x} - \frac{1 - \log{\left(x \right)}}{x^{2}} - \frac{1}{x^{2}}

  2. Now simplify:

    2x+log⁡(x)−2x2\frac{2 x + \log{\left(x \right)} - 2}{x^{2}}

The answer is:

2x+log⁡(x)−2x2\frac{2 x + \log{\left(x \right)} - 2}{x^{2}}

The graph
02468-8-6-4-2-1010-500500
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  2    2   log(x)
- -- + - + ------
   2   x      2  
  x          x
2x+log⁡(x)x2−2x2\frac{2}{x} + \frac{\log{\left(x \right)}}{x^{2}} - \frac{2}{x^{2}}
Simplify
The second derivative [src] 
     5   2*log(x)
-2 + - - --------
     x      x    
-----------------
         2       
        x
−2−2log⁡(x)x+5xx2\frac{-2 - \frac{2 \log{\left(x \right)}}{x} + \frac{5}{x}}{x^{2}}
Simplify
The third derivative [src] 
    17   6*log(x)
4 - -- + --------
    x       x    
-----------------
         3       
        x
4+6log⁡(x)x−17xx3\frac{4 + \frac{6 \log{\left(x \right)}}{x} - \frac{17}{x}}{x^{3}}
Simplify
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