Derivative of ln(x^2-x)

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   / 2    \
log\x  - x/

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

d /   / 2    \\
--\log\x  - x//
dx

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

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Detail solution

Let $u = x^{2} - x$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - x\right)$ :
1. Differentiate $x^{2} - x$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  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: $2 x - 1$
The result of the chain rule is:

$\frac{2 x - 1}{x^{2} - x}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

-1 + 2*x
--------
  2     
 x  - x

$$\frac{2 x - 1}{x^{2} - x}$$

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

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Derivative of ln(x^2-x)

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