Mister Exam

Lang:

Other calculators

How to use it?
Derivative of:
Derivative of 5*x
Derivative of (1-x^2)^(1/2)
Derivative of x-5
Derivative of x^(3*x)
Identical expressions
y=(x²- one)(x+ one)/x²-2x+ one
y equally (x² minus 1)(x plus 1) divide by x² minus 2x plus 1
y equally (x² minus one)(x plus one) divide by x² minus 2x plus one
y=x²-1x+1/x²-2x+1
y=(x²-1)(x+1) divide by x²-2x+1
Similar expressions
y=(x²+1)(x+1)/x²-2x+1
y=(x²-1)(x+1)/x²+2x+1
y=(x²-1)(x-1)/x²-2x+1
y=(x²-1)(x+1)/x²-2x-1

The derivative of the function/ y=(x²-1)(x+1)/x²-2x+1

Derivative of y=(x²-1)(x+1)/x²-2x+1

The solution

You have entered [src]

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

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

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

Detail solution

Differentiate $\left(- 2 x + \frac{\left(x + 1\right) \left(x^{2} - 1\right)}{x^{2}}\right) + 1$ term by term:
1. Differentiate $- 2 x + \frac{\left(x + 1\right) \left(x^{2} - 1\right)}{x^{2}}$ term by term:
  1. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \left(x + 1\right) \left(x^{2} - 1\right)$ and $g{\left(x \right)} = x^{2}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = x + 1$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. Differentiate $x + 1$ term by term:
        
        The derivative of the constant $1$ is zero.
        
        Apply the power rule: $x$ goes to $1$
        
        The result is: $1$
      $g{\left(x \right)} = x^{2} - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. Differentiate $x^{2} - 1$ term by term:
        
        The derivative of the constant $-1$ is zero.
        
        Apply the power rule: $x^{2}$ goes to $2 x$
        
        The result is: $2 x$
      The result is: $x^{2} + 2 x \left(x + 1\right) - 1$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Apply the power rule: $x^{2}$ goes to $2 x$
    Now plug in to the quotient rule:
    
    $\frac{x^{2} \left(x^{2} + 2 x \left(x + 1\right) - 1\right) - 2 x \left(x + 1\right) \left(x^{2} - 1\right)}{x^{4}}$
  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: $-2 + \frac{x^{2} \left(x^{2} + 2 x \left(x + 1\right) - 1\right) - 2 x \left(x + 1\right) \left(x^{2} - 1\right)}{x^{4}}$
2. The derivative of the constant $1$ is zero.
The result is: $-2 + \frac{x^{2} \left(x^{2} + 2 x \left(x + 1\right) - 1\right) - 2 x \left(x + 1\right) \left(x^{2} - 1\right)}{x^{4}}$
Now simplify:

$\frac{- x^{3} + x + 2}{x^{3}}$

The answer is:

$\frac{- x^{3} + x + 2}{x^{3}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                     /      2              \     /      2\                           /      2\\
  |     4*(1 + 3*x)   3*\-1 + x  + 2*x*(1 + x)/   6*\-1 + x /   10*(1 + x)   12*(1 + x)*\-1 + x /|
2*|-1 - ----------- + ------------------------- + ----------- + ---------- - --------------------|
  |          x                     2                    2           x                  3         |
  \                               x                    x                              x          /
--------------------------------------------------------------------------------------------------
                                                 2                                                
                                                x

$$\frac{2 \left(-1 + \frac{10 \left(x + 1\right)}{x} - \frac{4 \left(3 x + 1\right)}{x} + \frac{6 \left(x^{2} - 1\right)}{x^{2}} + \frac{3 \left(x^{2} + 2 x \left(x + 1\right) - 1\right)}{x^{2}} - \frac{12 \left(x + 1\right) \left(x^{2} - 1\right)}{x^{3}}\right)}{x^{2}}$$

Simplify

The graph

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of y=(x²-1)(x+1)/x²-2x+1

The graph:

Piecewise:

The solution