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Identical expressions
(ln(- three x+ two))/((x^3)-5x)
(ln( minus 3x plus 2)) divide by ((x cubed ) minus 5x)
(ln( minus three x plus two)) divide by ((x cubed ) minus 5x)
(ln(-3x+2))/((x3)-5x)
ln-3x+2/x3-5x
(ln(-3x+2))/((x³)-5x)
(ln(-3x+2))/((x to the power of 3)-5x)
ln-3x+2/x^3-5x
(ln(-3x+2)) divide by ((x^3)-5x)
Similar expressions
(ln(3x+2))/((x^3)-5x)
(ln(-3x+2))/((x^3)+5x)
(ln(-3x-2))/((x^3)-5x)

The derivative of the function/ (ln(-3x+2))/((x^3)-5x)

Derivative of (ln(-3x+2))/((x^3)-5x)

The solution

You have entered [src]

log(-3*x + 2)
-------------
    3        
   x  - 5*x

$$\frac{\log{\left(2 - 3 x \right)}}{x^{3} - 5 x}$$

log(-3*x + 2)/(x^3 - 5*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(2 - 3 x \right)}$ and $g{\left(x \right)} = x^{3} - 5 x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Let $u = 2 - 3 x$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 - 3 x\right)$ :
  1. Differentiate $2 - 3 x$ term by term:
    1. 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: $-3$
    2. The derivative of the constant $2$ is zero.
    The result is: $-3$
  The result of the chain rule is:
  
  $- \frac{3}{2 - 3 x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{3} - 5 x$ term by term:
  1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  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: $-5$
  The result is: $3 x^{2} - 5$
Now plug in to the quotient rule:

$\frac{- \left(3 x^{2} - 5\right) \log{\left(2 - 3 x \right)} - \frac{3 \left(x^{3} - 5 x\right)}{2 - 3 x}}{\left(x^{3} - 5 x\right)^{2}}$
Now simplify:

$\frac{3 x^{3} - 15 x - \left(3 x - 2\right) \left(3 x^{2} - 5\right) \log{\left(2 - 3 x \right)}}{x^{2} \left(3 x - 2\right) \left(x^{2} - 5\right)^{2}}$

The answer is:

$\frac{3 x^{3} - 15 x - \left(3 x - 2\right) \left(3 x^{2} - 5\right) \log{\left(2 - 3 x \right)}}{x^{2} \left(3 x - 2\right) \left(x^{2} - 5\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                          /       2\              
            3             \5 - 3*x /*log(-3*x + 2)
- --------------------- + ------------------------
  / 3      \                              2       
  \x  - 5*x/*(-3*x + 2)         / 3      \        
                                \x  - 5*x/

$$\frac{\left(5 - 3 x^{2}\right) \log{\left(2 - 3 x \right)}}{\left(x^{3} - 5 x\right)^{2}} - \frac{3}{\left(2 - 3 x\right) \left(x^{3} - 5 x\right)}$$

Simplify

The second derivative [src]

 /                /               2\                                      \ 
 |                |    /        2\ |                                      | 
 |                |    \-5 + 3*x / |                                      | 
 |              2*|3 - ------------|*log(2 - 3*x)                         | 
 |                |     2 /      2\|                      /        2\     | 
 |     9          \    x *\-5 + x //                    6*\-5 + 3*x /     | 
-|----------- + --------------------------------- + ----------------------| 
 |          2                      2                  /      2\           | 
 \(-2 + 3*x)                 -5 + x                 x*\-5 + x /*(-2 + 3*x)/ 
----------------------------------------------------------------------------
                                  /      2\                                 
                                x*\-5 + x /

$$- \frac{\frac{2 \left(3 - \frac{\left(3 x^{2} - 5\right)^{2}}{x^{2} \left(x^{2} - 5\right)}\right) \log{\left(2 - 3 x \right)}}{x^{2} - 5} + \frac{9}{\left(3 x - 2\right)^{2}} + \frac{6 \left(3 x^{2} - 5\right)}{x \left(3 x - 2\right) \left(x^{2} - 5\right)}}{x \left(x^{2} - 5\right)}$$

Simplify

The third derivative [src]

  /                                       /                                3\                                       \
  |                /               2\     |      /        2\    /        2\ |                                       |
  |                |    /        2\ |     |    6*\-5 + 3*x /    \-5 + 3*x / |                                       |
  |                |    \-5 + 3*x / |   2*|1 - ------------- + -------------|*log(2 - 3*x)                          |
  |              6*|3 - ------------|     |             2                  2|                                       |
  |                |     2 /      2\|     |       -5 + x        2 /      2\ |                       /        2\     |
  |     18         \    x *\-5 + x //     \                    x *\-5 + x / /                     9*\-5 + 3*x /     |
3*|----------- - -------------------- - -------------------------------------------------- + -----------------------|
  |          3   /      2\                                   /      2\                         /      2\           2|
  \(-2 + 3*x)    \-5 + x /*(-2 + 3*x)                      x*\-5 + x /                       x*\-5 + x /*(-2 + 3*x) /
---------------------------------------------------------------------------------------------------------------------
                                                       /      2\                                                     
                                                     x*\-5 + x /

$$\frac{3 \left(- \frac{6 \left(3 - \frac{\left(3 x^{2} - 5\right)^{2}}{x^{2} \left(x^{2} - 5\right)}\right)}{\left(3 x - 2\right) \left(x^{2} - 5\right)} + \frac{18}{\left(3 x - 2\right)^{3}} - \frac{2 \left(1 - \frac{6 \left(3 x^{2} - 5\right)}{x^{2} - 5} + \frac{\left(3 x^{2} - 5\right)^{3}}{x^{2} \left(x^{2} - 5\right)^{2}}\right) \log{\left(2 - 3 x \right)}}{x \left(x^{2} - 5\right)} + \frac{9 \left(3 x^{2} - 5\right)}{x \left(3 x - 2\right)^{2} \left(x^{2} - 5\right)}\right)}{x \left(x^{2} - 5\right)}$$

Simplify

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Derivative of (ln(-3x+2))/((x^3)-5x)

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