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Identical expressions
sinx/(3x^ two - nine)
sinus of x divide by (3x squared minus 9)
sinus of x divide by (3x to the power of two minus nine)
sinx/(3x2-9)
sinx/3x2-9
sinx/(3x²-9)
sinx/(3x to the power of 2-9)
sinx/3x^2-9
sinx divide by (3x^2-9)
Similar expressions
sinx/(3x^2+9)

The derivative of the function/ sinx/(3x^2-9)

Derivative of sinx/(3x^2-9)

The solution

You have entered [src]

 sin(x) 
--------
   2    
3*x  - 9

$$\frac{\sin{\left(x \right)}}{3 x^{2} - 9}$$

sin(x)/(3*x^2 - 9)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = 3 x^{2} - 9$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. The derivative of sine is cosine:
  
  $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $3 x^{2} - 9$ 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: $6 x$
  The result is: $6 x$
Now plug in to the quotient rule:

$\frac{- 6 x \sin{\left(x \right)} + \left(3 x^{2} - 9\right) \cos{\left(x \right)}}{\left(3 x^{2} - 9\right)^{2}}$
Now simplify:

$\frac{- 2 x \sin{\left(x \right)} + \left(x^{2} - 3\right) \cos{\left(x \right)}}{3 \left(x^{2} - 3\right)^{2}}$

The answer is:

$\frac{- 2 x \sin{\left(x \right)} + \left(x^{2} - 3\right) \cos{\left(x \right)}}{3 \left(x^{2} - 3\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 cos(x)     6*x*sin(x)
-------- - -----------
   2                 2
3*x  - 9   /   2    \ 
           \3*x  - 9/

$$- \frac{6 x \sin{\left(x \right)}}{\left(3 x^{2} - 9\right)^{2}} + \frac{\cos{\left(x \right)}}{3 x^{2} - 9}$$

Simplify

The second derivative [src]

                         /          2 \       
                         |       4*x  |       
                       2*|-1 + -------|*sin(x)
                         |           2|       
          4*x*cos(x)     \     -3 + x /       
-sin(x) - ---------- + -----------------------
                 2                   2        
           -3 + x              -3 + x         
----------------------------------------------
                   /      2\                  
                 3*\-3 + x /

$$\frac{- \frac{4 x \cos{\left(x \right)}}{x^{2} - 3} - \sin{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 3} - 1\right) \sin{\left(x \right)}}{x^{2} - 3}}{3 \left(x^{2} - 3\right)}$$

Simplify

The third derivative [src]

                          /          2 \              /          2 \       
                          |       4*x  |              |       2*x  |       
                        2*|-1 + -------|*cos(x)   8*x*|-1 + -------|*sin(x)
                          |           2|              |           2|       
  cos(x)   2*x*sin(x)     \     -3 + x /              \     -3 + x /       
- ------ + ---------- + ----------------------- - -------------------------
    3             2                   2                            2       
            -3 + x              -3 + x                    /      2\        
                                                          \-3 + x /        
---------------------------------------------------------------------------
                                        2                                  
                                  -3 + x

$$\frac{\frac{2 x \sin{\left(x \right)}}{x^{2} - 3} - \frac{8 x \left(\frac{2 x^{2}}{x^{2} - 3} - 1\right) \sin{\left(x \right)}}{\left(x^{2} - 3\right)^{2}} - \frac{\cos{\left(x \right)}}{3} + \frac{2 \left(\frac{4 x^{2}}{x^{2} - 3} - 1\right) \cos{\left(x \right)}}{x^{2} - 3}}{x^{2} - 3}$$

Simplify

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Derivative of sinx/(3x^2-9)

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The solution