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Similar expressions
log((2*x-1)/(x-2))
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The derivative of the function/ log((2*x-1)/(x+2))

Derivative of log((2*x-1)/(x+2))

The solution

You have entered [src]

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

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

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

Detail solution

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

$\frac{5 \left(x + 2\right)}{\left(x + 2\right)^{2} \left(2 x - 1\right)}$
Now simplify:

$\frac{5}{\left(x + 2\right) \left(2 x - 1\right)}$

The answer is:

$\frac{5}{\left(x + 2\right) \left(2 x - 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/    -1 + 2*x\ /    1        2    \
|2 - --------|*|- ----- - --------|
\     2 + x  / \  2 + x   -1 + 2*x/
-----------------------------------
              -1 + 2*x

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

Simplify

The third derivative [src]

  /    -1 + 2*x\ /   1            4                2         \
2*|2 - --------|*|-------- + ----------- + ------------------|
  \     2 + x  / |       2             2   (-1 + 2*x)*(2 + x)|
                 \(2 + x)    (-1 + 2*x)                      /
--------------------------------------------------------------
                           -1 + 2*x

$$\frac{2 \left(2 - \frac{2 x - 1}{x + 2}\right) \left(\frac{4}{\left(2 x - 1\right)^{2}} + \frac{2}{\left(x + 2\right) \left(2 x - 1\right)} + \frac{1}{\left(x + 2\right)^{2}}\right)}{2 x - 1}$$

Simplify

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Derivative of log((2*x-1)/(x+2))

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The solution