Derivative of (2-x)/log(x)

You have entered [src]

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

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

(2 - x)/log(x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 - x$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 - x$ term by term:
  1. The derivative of the constant $2$ is zero.
  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: $-1$
  The result is: $-1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
Now plug in to the quotient rule:

$\frac{- \log{\left(x \right)} - \frac{2 - x}{x}}{\log{\left(x \right)}^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

    1        2 - x  
- ------ - ---------
  log(x)        2   
           x*log (x)

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

Simplify

The second derivative [src]

    /      2   \         
    |1 + ------|*(-2 + x)
    \    log(x)/         
2 - ---------------------
              x          
-------------------------
             2           
        x*log (x)

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

Simplify

The third derivative [src]

                         /      3         3   \
              2*(-2 + x)*|1 + ------ + -------|
                         |    log(x)      2   |
       6                 \             log (x)/
-3 - ------ + ---------------------------------
     log(x)                   x                
-----------------------------------------------
                    2    2                     
                   x *log (x)

$$\frac{-3 - \frac{6}{\log{\left(x \right)}} + \frac{2 \left(x - 2\right) \left(1 + \frac{3}{\log{\left(x \right)}} + \frac{3}{\log{\left(x \right)}^{2}}\right)}{x}}{x^{2} \log{\left(x \right)}^{2}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of (2-x)/log(x)

The graph:

Piecewise:

The solution