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Derivative of:
Derivative of 5*x^2-2/sqrt(x)+sin(pi/4)
Derivative of 2*cos(3*x)
Derivative of (1+x)
Derivative of x*lnx
Identical expressions
(x+ one)/(x- one)^ two
(x plus 1) divide by (x minus 1) squared
(x plus one) divide by (x minus one) to the power of two
(x+1)/(x-1)2
x+1/x-12
(x+1)/(x-1)²
(x+1)/(x-1) to the power of 2
x+1/x-1^2
(x+1) divide by (x-1)^2
Similar expressions
(x+1)/(x+1)^2
(x-1)/(x-1)^2

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

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

The solution

You have entered [src]

 x + 1  
--------
       2
(x - 1)

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

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

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /    4*(1 + x)\
6*|3 - ---------|
  \      -1 + x /
-----------------
            4    
    (-1 + x)

$$\frac{6 \left(3 - \frac{4 \left(x + 1\right)}{x - 1}\right)}{\left(x - 1\right)^{4}}$$

Simplify

The graph

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