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Derivative of y=cos5x
Identical expressions
(sqrt(two *x+ five)- five)/(x^ two - one)
( square root of (2 multiply by x plus 5) minus 5) divide by (x squared minus 1)
( square root of (two multiply by x plus five) minus five) divide by (x to the power of two minus one)
(√(2*x+5)-5)/(x^2-1)
(sqrt(2*x+5)-5)/(x2-1)
sqrt2*x+5-5/x2-1
(sqrt(2*x+5)-5)/(x²-1)
(sqrt(2*x+5)-5)/(x to the power of 2-1)
(sqrt(2x+5)-5)/(x^2-1)
(sqrt(2x+5)-5)/(x2-1)
sqrt2x+5-5/x2-1
sqrt2x+5-5/x^2-1
(sqrt(2*x+5)-5) divide by (x^2-1)
Similar expressions
(sqrt(2*x+5)-5)/(x^2+1)
(sqrt(2*x+5)+5)/(x^2-1)
(sqrt(2*x-5)-5)/(x^2-1)

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

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

The solution

You have entered [src]

  _________    
\/ 2*x + 5  - 5
---------------
      2        
     x  - 1

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

  /  _________    \
d |\/ 2*x + 5  - 5|
--|---------------|
dx|      2        |
  \     x  - 1    /

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

Transform

Detail solution

Apply the quotient rule, which is:

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

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

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

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

$\frac{- 3 x^{2} + 10 x \sqrt{2 x + 5} - 10 x - 1}{\sqrt{2 x + 5} \left(x^{4} - 2 x^{2} + 1\right)}$

The answer is:

$\frac{- 3 x^{2} + 10 x \sqrt{2 x + 5} - 10 x - 1}{\sqrt{2 x + 5} \left(x^{4} - 2 x^{2} + 1\right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                           /  _________    \
         1             2*x*\\/ 2*x + 5  - 5/
-------------------- - ---------------------
/ 2    \   _________                 2      
\x  - 1/*\/ 2*x + 5          / 2    \       
                             \x  - 1/

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                             /          2 \         /          2 \                   \
  |                                             |       4*x  |         |       2*x  | /       _________\|
  |                                           2*|-1 + -------|     8*x*|-1 + -------|*\-5 + \/ 5 + 2*x /|
  |                                             |           2|         |           2|                   |
  |     1                  2*x                  \     -1 + x /         \     -1 + x /                   |
3*|------------ + ---------------------- + --------------------- - -------------------------------------|
  |         5/2   /      2\          3/2   /      2\   _________                          2             |
  |(5 + 2*x)      \-1 + x /*(5 + 2*x)      \-1 + x /*\/ 5 + 2*x                  /      2\              |
  \                                                                              \-1 + x /              /
---------------------------------------------------------------------------------------------------------
                                                       2                                                 
                                                 -1 + x

$$\frac{3 \left(- \frac{8 x \left(\frac{2 x^{2}}{x^{2} - 1} - 1\right) \left(\sqrt{2 x + 5} - 5\right)}{\left(x^{2} - 1\right)^{2}} + \frac{2 x}{\left(2 x + 5\right)^{\frac{3}{2}} \left(x^{2} - 1\right)} + \frac{2 \cdot \left(\frac{4 x^{2}}{x^{2} - 1} - 1\right)}{\sqrt{2 x + 5} \left(x^{2} - 1\right)} + \frac{1}{\left(2 x + 5\right)^{\frac{5}{2}}}\right)}{x^{2} - 1}$$

Simplify

The graph

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

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