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Derivative of x^2*cos(2*x)
Derivative of ((x-1)/(x+1))^4
Derivative of (x-1)/(2*x+3)
Derivative of sqrt(x)^2-2*x
Identical expressions
((x- one)/(x+ one))^ four
((x minus 1) divide by (x plus 1)) to the power of 4
((x minus one) divide by (x plus one)) to the power of four
((x-1)/(x+1))4
x-1/x+14
((x-1)/(x+1))⁴
x-1/x+1^4
((x-1) divide by (x+1))^4
Similar expressions
((x+1)/(x+1))^4
((x-1)/(x-1))^4

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

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

The solution

You have entered [src]

       4
/x - 1\ 
|-----| 
\x + 1/

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

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

Detail solution

Let $u = \frac{x - 1}{x + 1}$ .
Apply the power rule: $u^{4}$ goes to $4 u^{3}$
Then, apply the chain rule. Multiply by $\frac{d}{d x} \frac{x - 1}{x + 1}$ :
1. 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 $g{\left(x \right)} = x + 1$ .
  
  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 $x + 1$ 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:
  
  $\frac{2}{\left(x + 1\right)^{2}}$
The result of the chain rule is:

$\frac{8 \left(x - 1\right)^{3}}{\left(x + 1\right)^{2} \left(x + 1\right)^{3}}$
Now simplify:

$\frac{8 \left(x - 1\right)^{3}}{\left(x + 1\right)^{5}}$

The answer is:

$\frac{8 \left(x - 1\right)^{3}}{\left(x + 1\right)^{5}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

          2 /     -1 + x\ /     5*(-1 + x)\
4*(-1 + x) *|-1 + ------|*|-3 + ----------|
            \     1 + x / \       1 + x   /
-------------------------------------------
                         4                 
                  (1 + x)

$$\frac{4 \left(x - 1\right)^{2} \left(\frac{x - 1}{x + 1} - 1\right) \left(\frac{5 \left(x - 1\right)}{x + 1} - 3\right)}{\left(x + 1\right)^{4}}$$

Simplify

The third derivative [src]

                         /                                             /     -1 + x\\
                         |                2                 2*(-1 + x)*|-1 + ------||
           /     -1 + x\ |     13*(-1 + x)    13*(-1 + x)              \     1 + x /|
8*(-1 + x)*|-1 + ------|*|-3 - ------------ + ----------- - ------------------------|
           \     1 + x / |              2        1 + x               1 + x          |
                         \       (1 + x)                                            /
-------------------------------------------------------------------------------------
                                              4                                      
                                       (1 + x)

$$\frac{8 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right) \left(- \frac{13 \left(x - 1\right)^{2}}{\left(x + 1\right)^{2}} - \frac{2 \left(x - 1\right) \left(\frac{x - 1}{x + 1} - 1\right)}{x + 1} + \frac{13 \left(x - 1\right)}{x + 1} - 3\right)}{\left(x + 1\right)^{4}}$$

Simplify

The graph

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

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Piecewise:

The solution