Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of -x+1
Derivative of sqrt((1+x)/(1-x))
Derivative of sin(t)*cos(t)
Derivative of logax
Graphing y =:
sqrt((1+x)/(1-x))
Limit of the function:
sqrt((1+x)/(1-x))
Integral of d{x}:
sqrt((1+x)/(1-x))
Identical expressions
sqrt((one +x)/(one -x))
square root of ((1 plus x) divide by (1 minus x))
square root of ((one plus x) divide by (one minus x))
√((1+x)/(1-x))
sqrt1+x/1-x
sqrt((1+x) divide by (1-x))
Similar expressions
sqrt((1-x)/(1-x))
(sqrt(1+x)/(1-x))
sqrt((1+x)/(1+x))

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

Derivative of sqrt((1+x)/(1-x))

The solution

You have entered [src]

    _______
   / 1 + x 
  /  ----- 
\/   1 - x

$$\sqrt{\frac{x + 1}{1 - x}}$$

  /    _______\
d |   / 1 + x |
--|  /  ----- |
dx\\/   1 - x /

$$\frac{d}{d x} \sqrt{\frac{x + 1}{1 - x}}$$

Transform

Detail solution

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

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of sqrt((1+x)/(1-x))

The graph:

Piecewise:

The solution