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cosx/(x^2(x+1))

The derivative of the function/ cosx/(x^2(x-1))

Derivative of cosx/(x^2(x-1))

The solution

You have entered [src]

  cos(x)  
----------
 2        
x *(x - 1)

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

cos(x)/((x^2*(x - 1)))

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 $g{\left(x \right)} = x^{2} \left(x - 1\right)$ .

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. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x^{2}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  $g{\left(x \right)} = x - 1$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x - 1$ term by term:
    1. The derivative of the constant $-1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result is: $x^{2} + 2 x \left(x - 1\right)$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

  /                                                                                                           /  1      2\                               /  1      2\                /  1      2\               \                                                                                                                    
  |                                                                                                (-2 + 3*x)*|------ + -|                  2*(-1 + 3*x)*|------ + -|   2*(-2 + 3*x)*|------ + -|               |                                                                                                                    
  |6   12*(-1 + 3*x)                /    1       3        2     \   3*(-2 + 3*x)   10*(-2 + 3*x)              \-1 + x   x/   6*(-1 + 3*x)                \-1 + x   x/                \-1 + x   x/   8*(-2 + 3*x)|                                                                                                                    
  |- - ------------- + 2*(-2 + 3*x)*|--------- + -- + ----------| + ------------ + ------------- + ----------------------- - ------------ - ------------------------- + ------------------------- + ------------|*cos(x)     /-2 + 3*x              /  1      2\   2*(-1 + 3*x)   2*(-2 + 3*x)\                                      
  |x          2                     |        2    2   x*(-1 + x)|            2            2                 -1 + x            x*(-1 + x)                x                           x                x*(-1 + x) |          3*|-------- + (-2 + 3*x)*|------ + -| - ------------ + ------------|*sin(x)                               
  \          x                      \(-1 + x)    x              /    (-1 + x)            x                                                                                                                      /            \ -1 + x               \-1 + x   x/        x              x      /          3*(-2 + 3*x)*cos(x)         
- ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------- + ------------------- + sin(x)
                                                                                                        x*(-1 + x)                                                                                                                                          x*(-1 + x)                                        x*(-1 + x)             
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                                                                              2                                                                                                                                                                      
                                                                                                                                                             x *(-1 + x)

$$\frac{\sin{\left(x \right)} + \frac{3 \left(3 x - 2\right) \cos{\left(x \right)}}{x \left(x - 1\right)} - \frac{3 \left(\left(3 x - 2\right) \left(\frac{1}{x - 1} + \frac{2}{x}\right) + \frac{3 x - 2}{x - 1} + \frac{2 \left(3 x - 2\right)}{x} - \frac{2 \left(3 x - 1\right)}{x}\right) \sin{\left(x \right)}}{x \left(x - 1\right)} - \frac{\left(2 \left(3 x - 2\right) \left(\frac{1}{\left(x - 1\right)^{2}} + \frac{2}{x \left(x - 1\right)} + \frac{3}{x^{2}}\right) + \frac{\left(3 x - 2\right) \left(\frac{1}{x - 1} + \frac{2}{x}\right)}{x - 1} + \frac{3 \left(3 x - 2\right)}{\left(x - 1\right)^{2}} + \frac{2 \left(3 x - 2\right) \left(\frac{1}{x - 1} + \frac{2}{x}\right)}{x} - \frac{2 \left(3 x - 1\right) \left(\frac{1}{x - 1} + \frac{2}{x}\right)}{x} + \frac{6}{x} + \frac{8 \left(3 x - 2\right)}{x \left(x - 1\right)} - \frac{6 \left(3 x - 1\right)}{x \left(x - 1\right)} + \frac{10 \left(3 x - 2\right)}{x^{2}} - \frac{12 \left(3 x - 1\right)}{x^{2}}\right) \cos{\left(x \right)}}{x \left(x - 1\right)}}{x^{2} \left(x - 1\right)}$$

Simplify

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Derivative of cosx/(x^2(x-1))

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