Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of arctg(1/x)
Derivative of e^cos(x)
Derivative of ln(x+3)^3
Derivative of cot(1/x)
Graphing y =:
(x^2-x+1)/(x-1)
Identical expressions
(x^ two -x+ one)/(x- one)
(x squared minus x plus 1) divide by (x minus 1)
(x to the power of two minus x plus one) divide by (x minus one)
(x2-x+1)/(x-1)
x2-x+1/x-1
(x²-x+1)/(x-1)
(x to the power of 2-x+1)/(x-1)
x^2-x+1/x-1
(x^2-x+1) divide by (x-1)
Similar expressions
(x^2-x-1)/(x-1)
(x^2-x+1)/(x+1)
(x^2+x+1)/(x-1)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} - x + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. Apply the power rule: $x^{2}$ goes to $2 x$
  3. 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: $-1$
  The result is: $2 x - 1$
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:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

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

The graph:

Piecewise:

The solution