Derivative of ln((5x-1)/(3x+2))

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   /5*x - 1\
log|-------|
   \3*x + 2/

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

log((5*x - 1)/(3*x + 2))

Detail solution

Let $u = \frac{5 x - 1}{3 x + 2}$ .
The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{5 x - 1}{3 x + 2}$ :
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 5 x - 1$ and $g{\left(x \right)} = 3 x + 2$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Differentiate $5 x - 1$ 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: $5$
    The result is: $5$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $3 x + 2$ term by term:
    1. The derivative of the constant $2$ 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: $3$
    The result is: $3$
  Now plug in to the quotient rule:
  
  $\frac{13}{\left(3 x + 2\right)^{2}}$
The result of the chain rule is:

$\frac{13 \left(3 x + 2\right)}{\left(3 x + 2\right)^{2} \left(5 x - 1\right)}$
Now simplify:

$\frac{13}{\left(3 x + 2\right) \left(5 x - 1\right)}$

The answer is:

$\frac{13}{\left(3 x + 2\right) \left(5 x - 1\right)}$

The graph

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The first derivative [src]

          /   5      3*(5*x - 1)\
(3*x + 2)*|------- - -----------|
          |3*x + 2             2|
          \           (3*x + 2) /
---------------------------------
             5*x - 1

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

Simplify

The second derivative [src]

/    3*(-1 + 5*x)\ /     5          3   \
|5 - ------------|*|- -------- - -------|
\      2 + 3*x   / \  -1 + 5*x   2 + 3*x/
-----------------------------------------
                 -1 + 5*x

$$\frac{\left(5 - \frac{3 \left(5 x - 1\right)}{3 x + 2}\right) \left(- \frac{5}{5 x - 1} - \frac{3}{3 x + 2}\right)}{5 x - 1}$$

Simplify

The third derivative [src]

  /    3*(-1 + 5*x)\ /    9             25                15         \
2*|5 - ------------|*|---------- + ----------- + --------------------|
  \      2 + 3*x   / |         2             2   (-1 + 5*x)*(2 + 3*x)|
                     \(2 + 3*x)    (-1 + 5*x)                        /
----------------------------------------------------------------------
                               -1 + 5*x

$$\frac{2 \left(5 - \frac{3 \left(5 x - 1\right)}{3 x + 2}\right) \left(\frac{25}{\left(5 x - 1\right)^{2}} + \frac{15}{\left(3 x + 2\right) \left(5 x - 1\right)} + \frac{9}{\left(3 x + 2\right)^{2}}\right)}{5 x - 1}$$

Simplify

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Derivative of ln((5x-1)/(3x+2))

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The solution