Mister Exam

Lang:

The derivative of the function/ lnx/x-1

Derivative of lnx/x-1

The solution

You have entered [src]

log(x)    
------ - 1
  x

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

log(x)/x - 1

Detail solution

Differentiate $-1 + \frac{\log{\left(x \right)}}{x}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ 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:
  
  $\frac{1 - \log{\left(x \right)}}{x^{2}}$
2. The derivative of the constant $-1$ is zero.
The result is: $\frac{1 - \log{\left(x \right)}}{x^{2}}$

The answer is:

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

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

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

Simplify

The third derivative [src]

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

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

Simplify