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The derivative of the function/ -lnx/x

Derivative of -lnx/x

The solution

You have entered [src]

-log(x) 
--------
   x

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

(-log(x))/x

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = - \log{\left(x \right)}$ and $g{\left(x \right)} = x$ .

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. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  So, the result is: $- \frac{1}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

$\frac{\log{\left(x \right)} - 1}{x^{2}}$

The answer is:

$\frac{\log{\left(x \right)} - 1}{x^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  1    log(x)
- -- + ------
   2      2  
  x      x

$$\frac{\log{\left(x \right)}}{x^{2}} - \frac{1}{x^{2}}$$

Simplify

The second derivative [src]

3 - 2*log(x)
------------
      3     
     x

$$\frac{3 - 2 \log{\left(x \right)}}{x^{3}}$$

Simplify

The third derivative [src]

-11 + 6*log(x)
--------------
       4      
      x

$$\frac{6 \log{\left(x \right)} - 11}{x^{4}}$$

Simplify