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((5x^ two)-x+ four)/(x- one)
((5x squared ) minus x plus 4) divide by (x minus 1)
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((5x^2)-x-4)/(x-1)
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The derivative of the function/ ((5x^2)-x+4)/(x-1)

Derivative of ((5x^2)-x+4)/(x-1)

The solution

You have entered [src]

   2        
5*x  - x + 4
------------
   x - 1

$$\frac{\left(5 x^{2} - x\right) + 4}{x - 1}$$

(5*x^2 - x + 4)/(x - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 5 x^{2} - x + 4$ and $g{\left(x \right)} = x - 1$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $5 x^{2} - x + 4$ term by term:
  1. The derivative of the constant $4$ 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: $-1$
  3. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    So, the result is: $10 x$
  The result is: $10 x - 1$
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:

$\frac{- 5 x^{2} + x + \left(x - 1\right) \left(10 x - 1\right) - 4}{\left(x - 1\right)^{2}}$
Now simplify:

$\frac{5 x^{2} - 10 x - 3}{x^{2} - 2 x + 1}$

The answer is:

$\frac{5 x^{2} - 10 x - 3}{x^{2} - 2 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               2        
-1 + 10*x   5*x  - x + 4
--------- - ------------
  x - 1              2  
              (x - 1)

$$\frac{10 x - 1}{x - 1} - \frac{\left(5 x^{2} - x\right) + 4}{\left(x - 1\right)^{2}}$$

Simplify

The second derivative [src]

  /               2            \
  |    4 - x + 5*x    -1 + 10*x|
2*|5 + ------------ - ---------|
  |             2       -1 + x |
  \     (-1 + x)               /
--------------------------------
             -1 + x

$$\frac{2 \left(5 - \frac{10 x - 1}{x - 1} + \frac{5 x^{2} - x + 4}{\left(x - 1\right)^{2}}\right)}{x - 1}$$

Simplify

The third derivative [src]

  /                            2\
  |     -1 + 10*x   4 - x + 5*x |
6*|-5 + --------- - ------------|
  |       -1 + x             2  |
  \                  (-1 + x)   /
---------------------------------
                    2            
            (-1 + x)

$$\frac{6 \left(-5 + \frac{10 x - 1}{x - 1} - \frac{5 x^{2} - x + 4}{\left(x - 1\right)^{2}}\right)}{\left(x - 1\right)^{2}}$$

Simplify

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Derivative of ((5x^2)-x+4)/(x-1)

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The solution