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Equation:
(-x)/(x^2+225)
Identical expressions
(-x)/(x^ two + two hundred and twenty-five)
( minus x) divide by (x squared plus 225)
( minus x) divide by (x to the power of two plus two hundred and twenty minus five)
(-x)/(x2+225)
-x/x2+225
(-x)/(x²+225)
(-x)/(x to the power of 2+225)
-x/x^2+225
(-x) divide by (x^2+225)
Similar expressions
(x)/(x^2+225)
(-x)/(x^2-225)

The derivative of the function/ (-x)/(x^2+225)

Derivative of (-x)/(x^2+225)

The solution

You have entered [src]

  -x    
--------
 2      
x  + 225

$$\frac{\left(-1\right) x}{x^{2} + 225}$$

(-x)/(x^2 + 225)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = - x$ and $g{\left(x \right)} = x^{2} + 225$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. 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: $-1$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $x^{2} + 225$ term by term:
  1. The derivative of the constant $225$ 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:

$\frac{x^{2} - 225}{\left(x^{2} + 225\right)^{2}}$

The answer is:

$\frac{x^{2} - 225}{\left(x^{2} + 225\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

                    2   
     1           2*x    
- -------- + -----------
   2                   2
  x  + 225   / 2      \ 
             \x  + 225/

$$\frac{2 x^{2}}{\left(x^{2} + 225\right)^{2}} - \frac{1}{x^{2} + 225}$$

Simplify

The second derivative [src]

    /         2  \
    |      4*x   |
2*x*|3 - --------|
    |           2|
    \    225 + x /
------------------
             2    
   /       2\     
   \225 + x /

$$\frac{2 x \left(- \frac{4 x^{2}}{x^{2} + 225} + 3\right)}{\left(x^{2} + 225\right)^{2}}$$

Simplify

The third derivative [src]

  /                    /          2  \\
  |                  2 |       2*x   ||
  |               4*x *|-1 + --------||
  |         2          |            2||
  |      4*x           \     225 + x /|
6*|1 - -------- + --------------------|
  |           2                2      |
  \    225 + x          225 + x       /
---------------------------------------
                        2              
              /       2\               
              \225 + x /

$$\frac{6 \left(\frac{4 x^{2} \left(\frac{2 x^{2}}{x^{2} + 225} - 1\right)}{x^{2} + 225} - \frac{4 x^{2}}{x^{2} + 225} + 1\right)}{\left(x^{2} + 225\right)^{2}}$$

Simplify

The graph

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Derivative of (-x)/(x^2+225)

The graph:

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The solution