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Identical expressions
(x^ two + two *x+ one)/(x*(- two)+ four)
(x squared plus 2 multiply by x plus 1) divide by (x multiply by ( minus 2) plus 4)
(x to the power of two plus two multiply by x plus one) divide by (x multiply by ( minus two) plus four)
(x2+2*x+1)/(x*(-2)+4)
x2+2*x+1/x*-2+4
(x²+2*x+1)/(x*(-2)+4)
(x to the power of 2+2*x+1)/(x*(-2)+4)
(x^2+2x+1)/(x(-2)+4)
(x2+2x+1)/(x(-2)+4)
x2+2x+1/x-2+4
x^2+2x+1/x-2+4
(x^2+2*x+1) divide by (x*(-2)+4)
Similar expressions
(x^2+2*x-1)/(x*(-2)+4)
(x^2+2*x+1)/(x*(2)+4)
(x^2-2*x+1)/(x*(-2)+4)
(x^2+2*x+1)/(x*(-2)-4)

The derivative of the function/ (x^2+2*x+1)/(x*(-2)+4)

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + 2 x + 1$ 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: $2$
  The result is: $2 x + 2$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $4 - 2 x$ 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: $-2$
  The result is: $-2$
Now plug in to the quotient rule:

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

$\frac{- x^{2} + 4 x + 5}{2 x^{2} - 8 x + 8}$

The answer is:

$\frac{- x^{2} + 4 x + 5}{2 x^{2} - 8 x + 8}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

             2                  
  1     1 + x  + 2*x   2*(1 + x)
----- - ------------ + ---------
2 - x            3             2
         (-2 + x)       (2 - x)

$$- \frac{x^{2} + 2 x + 1}{\left(x - 2\right)^{3}} + \frac{1}{2 - x} + \frac{2 \left(x + 1\right)}{\left(2 - x\right)^{2}}$$

Simplify

The third derivative [src]

  /                2                  \
  |   1       1 + x  + 2*x   2*(1 + x)|
3*|-------- + ------------ - ---------|
  |       2            4             3|
  \(2 - x)     (-2 + x)      (-2 + x) /

$$3 \left(- \frac{2 \left(x + 1\right)}{\left(x - 2\right)^{3}} + \frac{x^{2} + 2 x + 1}{\left(x - 2\right)^{4}} + \frac{1}{\left(2 - x\right)^{2}}\right)$$

Simplify

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

The graph:

Piecewise:

The solution

Other calculators

How to use it?

Identical expressions

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

The graph:

Piecewise:

The solution

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