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How do you in partial fractions?:
(1+x)/(4-x^2)
Limit of the function:
(1+x)/(4-x^2)
Identical expressions
(one +x)/(four -x^ two)
(1 plus x) divide by (4 minus x squared )
(one plus x) divide by (four minus x to the power of two)
(1+x)/(4-x2)
1+x/4-x2
(1+x)/(4-x²)
(1+x)/(4-x to the power of 2)
1+x/4-x^2
(1+x) divide by (4-x^2)
Similar expressions
(1-x)/(4-x^2)
(1+x)/(4+x^2)

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

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

The solution

You have entered [src]

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

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

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

Detail solution

Apply the quotient rule, which is:

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Identical expressions

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

The graph:

Piecewise:

The solution