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Derivative of:
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Identical expressions
cos(x)/(nine -x^ two)
co sinus of e of (x) divide by (9 minus x squared )
co sinus of e of (x) divide by (nine minus x to the power of two)
cos(x)/(9-x2)
cosx/9-x2
cos(x)/(9-x²)
cos(x)/(9-x to the power of 2)
cosx/9-x^2
cos(x) divide by (9-x^2)
Similar expressions
cos(x)/(9+x^2)
cosx/(9-x^2)

The derivative of the function/ cos(x)/(9-x^2)

Derivative of cos(x)/(9-x^2)

The solution

You have entered [src]

cos(x)
------
     2
9 - x

$$\frac{\cos{\left(x \right)}}{9 - x^{2}}$$

cos(x)/(9 - x^2)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \cos{\left(x \right)}$ and $g{\left(x \right)} = 9 - x^{2}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of cosine is negative sine:
  
  $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $9 - x^{2}$ term by term:
  1. The derivative of the constant $9$ 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: $- 2 x$
  The result is: $- 2 x$
Now plug in to the quotient rule:

$\frac{2 x \cos{\left(x \right)} - \left(9 - x^{2}\right) \sin{\left(x \right)}}{\left(9 - x^{2}\right)^{2}}$
Now simplify:

$\frac{2 x \cos{\left(x \right)} + \left(x^{2} - 9\right) \sin{\left(x \right)}}{\left(x^{2} - 9\right)^{2}}$

The answer is:

$\frac{2 x \cos{\left(x \right)} + \left(x^{2} - 9\right) \sin{\left(x \right)}}{\left(x^{2} - 9\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

  sin(x)   2*x*cos(x)
- ------ + ----------
       2           2 
  9 - x    /     2\  
           \9 - x /

$$\frac{2 x \cos{\left(x \right)}}{\left(9 - x^{2}\right)^{2}} - \frac{\sin{\left(x \right)}}{9 - x^{2}}$$

Simplify

The second derivative [src]

                 /          2 \                
                 |       4*x  |                
               2*|-1 + -------|*cos(x)         
                 |           2|                
  4*x*sin(x)     \     -9 + x /                
- ---------- - ----------------------- + cos(x)
         2                   2                 
   -9 + x              -9 + x                  
-----------------------------------------------
                          2                    
                    -9 + x

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

Simplify

The third derivative [src]

                         /          2 \               /          2 \       
                         |       4*x  |               |       2*x  |       
                       6*|-1 + -------|*sin(x)   24*x*|-1 + -------|*cos(x)
                         |           2|               |           2|       
          6*x*cos(x)     \     -9 + x /               \     -9 + x /       
-sin(x) - ---------- + ----------------------- + --------------------------
                 2                   2                            2        
           -9 + x              -9 + x                    /      2\         
                                                         \-9 + x /         
---------------------------------------------------------------------------
                                        2                                  
                                  -9 + x

$$\frac{- \frac{6 x \cos{\left(x \right)}}{x^{2} - 9} + \frac{24 x \left(\frac{2 x^{2}}{x^{2} - 9} - 1\right) \cos{\left(x \right)}}{\left(x^{2} - 9\right)^{2}} - \sin{\left(x \right)} + \frac{6 \left(\frac{4 x^{2}}{x^{2} - 9} - 1\right) \sin{\left(x \right)}}{x^{2} - 9}}{x^{2} - 9}$$

Simplify

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Identical expressions

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Derivative of cos(x)/(9-x^2)

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Piecewise:

The solution