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Identical expressions
(two x^ two - eighteen)/(x^2- twenty-five)
(2x squared minus 18) divide by (x squared minus 25)
(two x to the power of two minus eighteen) divide by (x squared minus twenty minus five)
(2x2-18)/(x2-25)
2x2-18/x2-25
(2x²-18)/(x²-25)
(2x to the power of 2-18)/(x to the power of 2-25)
2x^2-18/x^2-25
(2x^2-18) divide by (x^2-25)
Similar expressions
(2x^2+18)/(x^2-25)
(2x^2-18)/(x^2+25)

The derivative of the function/ (2x^2-18)/(x^2-25)

Derivative of (2x^2-18)/(x^2-25)

The solution

You have entered [src]

   2     
2*x  - 18
---------
  2      
 x  - 25

$$\frac{2 x^{2} - 18}{x^{2} - 25}$$

  /   2     \
d |2*x  - 18|
--|---------|
dx|  2      |
  \ x  - 25 /

$$\frac{d}{d x} \frac{2 x^{2} - 18}{x^{2} - 25}$$

Transform

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 2 x^{2} - 18$ and $g{\left(x \right)} = x^{2} - 25$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $2 x^{2} - 18$ term by term:
  1. The derivative of the constant $-18$ 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: $x^{2}$ goes to $2 x$
    So, the result is: $4 x$
  The result is: $4 x$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} - 25$ term by term:
  1. The derivative of the constant $-25$ 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:

$\frac{4 x \left(x^{2} - 25\right) - 2 x \left(2 x^{2} - 18\right)}{\left(x^{2} - 25\right)^{2}}$
Now simplify:

$- \frac{64 x}{\left(x^{2} - 25\right)^{2}}$

The answer is:

$- \frac{64 x}{\left(x^{2} - 25\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              /   2     \
  4*x     2*x*\2*x  - 18/
------- - ---------------
 2                    2  
x  - 25      / 2     \   
             \x  - 25/

$$\frac{4 x}{x^{2} - 25} - \frac{2 x \left(2 x^{2} - 18\right)}{\left(x^{2} - 25\right)^{2}}$$

Simplify

The second derivative [src]

  /               /          2  \          \
  |               |       4*x   | /      2\|
  |               |-1 + --------|*\-9 + x /|
  |         2     |            2|          |
  |      4*x      \     -25 + x /          |
4*|1 - -------- + -------------------------|
  |           2                   2        |
  \    -25 + x             -25 + x         /
--------------------------------------------
                         2                  
                  -25 + x

$$\frac{4 \left(- \frac{4 x^{2}}{x^{2} - 25} + \frac{\left(x^{2} - 9\right) \left(\frac{4 x^{2}}{x^{2} - 25} - 1\right)}{x^{2} - 25} + 1\right)}{x^{2} - 25}$$

Simplify

The third derivative [src]

     /                  /          2  \          \
     |                  |       2*x   | /      2\|
     |                2*|-1 + --------|*\-9 + x /|
     |          2       |            2|          |
     |       4*x        \     -25 + x /          |
24*x*|-2 + -------- - ---------------------------|
     |            2                    2         |
     \     -25 + x              -25 + x          /
--------------------------------------------------
                             2                    
                   /       2\                     
                   \-25 + x /

$$\frac{24 x \left(\frac{4 x^{2}}{x^{2} - 25} - \frac{2 \left(x^{2} - 9\right) \left(\frac{2 x^{2}}{x^{2} - 25} - 1\right)}{x^{2} - 25} - 2\right)}{\left(x^{2} - 25\right)^{2}}$$

Simplify

The graph

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Derivative of (2x^2-18)/(x^2-25)

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The solution