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The derivative of the function/ (5*x-x^2)/sqrt(x^2-1)

Derivative of (5*x-x^2)/sqrt(x^2-1)

The solution

You have entered [src]

         2 
  5*x - x  
-----------
   ________
  /  2     
\/  x  - 1

$$\frac{- x^{2} + 5 x}{\sqrt{x^{2} - 1}}$$

(5*x - x^2)/sqrt(x^2 - 1)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - x^{2} + 5 x$ and $g{\left(x \right)} = \sqrt{x^{2} - 1}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $- x^{2} + 5 x$ term by term:
  1. 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: $- 2 x$
  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: $5$
  The result is: $5 - 2 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x^{2} - 1$ .
2. Apply the power rule: $\sqrt{u}$ goes to $\frac{1}{2 \sqrt{u}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x^{2} - 1\right)$ :
  1. Differentiate $x^{2} - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x^{2}$ goes to $2 x$
    The result is: $2 x$
  The result of the chain rule is:
  
  $\frac{x}{\sqrt{x^{2} - 1}}$
Now plug in to the quotient rule:

$\frac{- \frac{x \left(- x^{2} + 5 x\right)}{\sqrt{x^{2} - 1}} + \left(5 - 2 x\right) \sqrt{x^{2} - 1}}{x^{2} - 1}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

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The first derivative [src]

                /       2\
  5 - 2*x     x*\5*x - x /
----------- - ------------
   ________           3/2 
  /  2        / 2    \    
\/  x  - 1    \x  - 1/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

The graph

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Derivative of (5*x-x^2)/sqrt(x^2-1)

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The solution