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The derivative of the function/ (x^2+1)/(x-1)

Derivative of (x^2+1)/(x-1)

The solution

You have entered [src]

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

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

  / 2    \
d |x  + 1|
--|------|
dx\x - 1 /

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

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = x^{2} + 1$ and $g{\left(x \right)} = x - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  The result is: $2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x - 1$ term by term:
  1. The derivative of the constant $-1$ is zero.
  2. Apply the power rule: $x$ goes to $1$
  The result is: $1$
Now plug in to the quotient rule:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /            2          \
  |       1 + x      2*x  |
6*|-1 - --------- + ------|
  |             2   -1 + x|
  \     (-1 + x)          /
---------------------------
                 2         
         (-1 + x)

$$\frac{6 \cdot \left(\frac{2 x}{x - 1} - 1 - \frac{x^{2} + 1}{\left(x - 1\right)^{2}}\right)}{\left(x - 1\right)^{2}}$$

Simplify

The graph

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Identical expressions

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Derivative of (x^2+1)/(x-1)

The graph:

Piecewise:

The solution