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Identical expressions
y=(x^ three -2x+ one)/ln(⁡x)
y equally (x cubed minus 2x plus 1) divide by ln(⁡x)
y equally (x to the power of three minus 2x plus one) divide by ln(⁡x)
y=(x3-2x+1)/ln(⁡x)
y=x3-2x+1/ln⁡x
y=(x³-2x+1)/ln(⁡x)
y=(x to the power of 3-2x+1)/ln(⁡x)
y=x^3-2x+1/ln⁡x
y=(x^3-2x+1) divide by ln(⁡x)
Similar expressions
y=(x^3-2x-1)/ln(⁡x)
y=(x^3+2x+1)/ln(⁡x)

The derivative of the function/ y=(x^3-2x+1)/ln(⁡x)

Derivative of y=(x^3-2x+1)/ln(⁡x)

The solution

You have entered [src]

 3          
x  - 2*x + 1
------------
   log(x)

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

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

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

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{3} - 2 x + 1$ and $g{\left(x \right)} = \log{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{3} - 2 x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
  3. 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: $-2$
  The result is: $3 x^{2} - 2$
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:

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

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

        2    3          
-2 + 3*x    x  - 2*x + 1
--------- - ------------
  log(x)          2     
             x*log (x)

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of y=(x^3-2x+1)/ln(⁡x)

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