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Derivative of x+5+6ln((x)/(x-1))

Function f() - derivative -N order at the point
v

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The solution

You have entered [src]
             /  x  \
x + 5 + 6*log|-----|
             \x - 1/
$$\left(x + 5\right) + 6 \log{\left(\frac{x}{x - 1} \right)}$$
x + 5 + 6*log(x/(x - 1))
Detail solution
  1. Differentiate term by term:

    1. Differentiate term by term:

      1. Apply the power rule: goes to

      2. The derivative of the constant is zero.

      The result is:

    2. The derivative of a constant times a function is the constant times the derivative of the function.

      1. Let .

      2. The derivative of is .

      3. Then, apply the chain rule. Multiply by :

        1. Apply the quotient rule, which is:

          and .

          To find :

          1. Apply the power rule: goes to

          To find :

          1. Differentiate term by term:

            1. The derivative of the constant is zero.

            2. Apply the power rule: goes to

            The result is:

          Now plug in to the quotient rule:

        The result of the chain rule is:

      So, the result is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
              /  1        x    \
    6*(x - 1)*|----- - --------|
              |x - 1          2|
              \        (x - 1) /
1 + ----------------------------
                 x              
$$1 + \frac{6 \left(x - 1\right) \left(- \frac{x}{\left(x - 1\right)^{2}} + \frac{1}{x - 1}\right)}{x}$$
The second derivative [src]
  /       x   \ /1     1   \
6*|-1 + ------|*|- + ------|
  \     -1 + x/ \x   -1 + x/
----------------------------
             x              
$$\frac{6 \left(\frac{x}{x - 1} - 1\right) \left(\frac{1}{x - 1} + \frac{1}{x}\right)}{x}$$
The third derivative [src]
   /       x   \ /  1        1           1     \
12*|-1 + ------|*|- -- - --------- - ----------|
   \     -1 + x/ |   2           2   x*(-1 + x)|
                 \  x    (-1 + x)              /
------------------------------------------------
                       x                        
$$\frac{12 \left(\frac{x}{x - 1} - 1\right) \left(- \frac{1}{\left(x - 1\right)^{2}} - \frac{1}{x \left(x - 1\right)} - \frac{1}{x^{2}}\right)}{x}$$