Derivative of ln((4x-3)/x)

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   /4*x - 3\
log|-------|
   \   x   /

$$\log{\left(\frac{4 x - 3}{x} \right)}$$

log((4*x - 3)/x)

Detail solution

Let $u = \frac{4 x - 3}{x}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{4 x - 3}{x}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 4 x - 3$ and $g{\left(x \right)} = x$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $4 x - 3$ term by term:
    1. The derivative of the constant $-3$ 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: $4$
    The result is: $4$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  Now plug in to the quotient rule:
  
  $\frac{3}{x^{2}}$
The result of the chain rule is:

$\frac{3}{x \left(4 x - 3\right)}$
Now simplify:

$\frac{3}{x \left(4 x - 3\right)}$

The answer is:

$\frac{3}{x \left(4 x - 3\right)}$

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The first derivative [src]

  /4   4*x - 3\
x*|- - -------|
  |x       2  |
  \       x   /
---------------
    4*x - 3

$$\frac{x \left(\frac{4}{x} - \frac{4 x - 3}{x^{2}}\right)}{4 x - 3}$$

Simplify

The second derivative [src]

/    -3 + 4*x\ /  1      4    \
|4 - --------|*|- - - --------|
\       x    / \  x   -3 + 4*x/
-------------------------------
            -3 + 4*x

$$\frac{\left(4 - \frac{4 x - 3}{x}\right) \left(- \frac{4}{4 x - 3} - \frac{1}{x}\right)}{4 x - 3}$$

Simplify

The third derivative [src]

  /    -3 + 4*x\ /1         16            4      \
2*|4 - --------|*|-- + ----------- + ------------|
  \       x    / | 2             2   x*(-3 + 4*x)|
                 \x    (-3 + 4*x)                /
--------------------------------------------------
                     -3 + 4*x

$$\frac{2 \left(4 - \frac{4 x - 3}{x}\right) \left(\frac{16}{\left(4 x - 3\right)^{2}} + \frac{4}{x \left(4 x - 3\right)} + \frac{1}{x^{2}}\right)}{4 x - 3}$$

Simplify

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Derivative of ln((4x-3)/x)

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