Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of sqrt(x^2+1)
Derivative of 1/sin(x)
Derivative of log(x+1)
Derivative of acos(x)
Identical expressions
y=x/(x- one)*(x- four)
y equally x divide by (x minus 1) multiply by (x minus 4)
y equally x divide by (x minus one) multiply by (x minus four)
y=x/(x-1)(x-4)
y=x/x-1x-4
y=x divide by (x-1)*(x-4)
Similar expressions
y=x/(x-1)*(x+4)
y=x/(x-1)(x-4)
x/((x-1)*(x-4))
y=x/((x-1)(x-4))
y=x/(x+1)*(x-4)

The derivative of the function/ y=x/(x-1)*(x-4)

Derivative of y=x/(x-1)*(x-4)

The solution

You have entered [src]

  x          
-----*(x - 4)
x - 1

$$\frac{x}{x - 1} \left(x - 4\right)$$

(x/(x - 1))*(x - 4)

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = x - 4$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 4$ term by term:
    1. The derivative of the constant $-4$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result is: $2 x - 4$
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:

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

$\frac{x^{2} - 2 x + 4}{x^{2} - 2 x + 1}$

The answer is:

$\frac{x^{2} - 2 x + 4}{x^{2} - 2 x + 1}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=x/(x-1)*(x-4)

The graph:

Piecewise:

The solution