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Identical expressions
(5x- three)/(sin^ three -x)
(5x minus 3) divide by ( sinus of cubed minus x)
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(5x-3)/(sin3-x)
5x-3/sin3-x
(5x-3)/(sin³-x)
(5x-3)/(sin to the power of 3-x)
5x-3/sin^3-x
(5x-3) divide by (sin^3-x)
Similar expressions
(5x-3)/(sin^3+x)
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The derivative of the function/ (5x-3)/(sin^3-x)

Derivative of (5x-3)/(sin^3-x)

The solution

You have entered [src]

  5*x - 3  
-----------
   3       
sin (x) - x

$$\frac{5 x - 3}{- x + \sin^{3}{\left(x \right)}}$$

(5*x - 3)/(sin(x)^3 - x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = 5 x - 3$ and $g{\left(x \right)} = - x + \sin^{3}{\left(x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $5 x - 3$ term by term:
  1. The derivative of the constant $-3$ 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: $x$ goes to $1$
    So, the result is: $5$
  The result is: $5$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Differentiate $- x + \sin^{3}{\left(x \right)}$ term by term:
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \sin{\left(x \right)}$ :
    1. The derivative of sine is cosine:
      
      $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
    The result of the chain rule is:
    
    $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
  4. 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: $-1$
  The result is: $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1$
Now plug in to the quotient rule:

$\frac{- 5 x - \left(5 x - 3\right) \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right) + 5 \sin^{3}{\left(x \right)}}{\left(- x + \sin^{3}{\left(x \right)}\right)^{2}}$
Now simplify:

$\frac{- 5 x + \left(3 - 5 x\right) \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right) + 5 \sin^{3}{\left(x \right)}}{\left(x - \sin^{3}{\left(x \right)}\right)^{2}}$

The answer is:

$\frac{- 5 x + \left(3 - 5 x\right) \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right) + 5 \sin^{3}{\left(x \right)}}{\left(x - \sin^{3}{\left(x \right)}\right)^{2}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              /         2          \          
     5        \1 - 3*sin (x)*cos(x)/*(5*x - 3)
----------- + --------------------------------
   3                                2         
sin (x) - x            /   3       \          
                       \sin (x) - x/

$$\frac{5}{- x + \sin^{3}{\left(x \right)}} + \frac{\left(5 x - 3\right) \left(- 3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 1\right)}{\left(- x + \sin^{3}{\left(x \right)}\right)^{2}}$$

Simplify

The second derivative [src]

                /                           2                                 \                    
                |    /          2          \                                  |                    
                |  2*\-1 + 3*sin (x)*cos(x)/      /   2           2   \       |         2          
10 + (-3 + 5*x)*|- -------------------------- + 3*\sin (x) - 2*cos (x)/*sin(x)| - 30*sin (x)*cos(x)
                |                3                                            |                    
                \         x - sin (x)                                         /                    
---------------------------------------------------------------------------------------------------
                                                        2                                          
                                           /       3   \                                           
                                           \x - sin (x)/

$$\frac{\left(5 x - 3\right) \left(3 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} - \frac{2 \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right)^{2}}{x - \sin^{3}{\left(x \right)}}\right) - 30 \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 10}{\left(x - \sin^{3}{\left(x \right)}\right)^{2}}$$

Simplify

The third derivative [src]

  /           /                                                            3                                                         \                             2                                  \
  |           |                                     /          2          \      /          2          \ /   2           2   \       |      /          2          \                                   |
  |           |/       2           2   \          2*\-1 + 3*sin (x)*cos(x)/    6*\-1 + 3*sin (x)*cos(x)/*\sin (x) - 2*cos (x)/*sin(x)|   10*\-1 + 3*sin (x)*cos(x)/       /   2           2   \       |
3*|(-3 + 5*x)*|\- 2*cos (x) + 7*sin (x)/*cos(x) - -------------------------- + ------------------------------------------------------| - --------------------------- + 15*\sin (x) - 2*cos (x)/*sin(x)|
  |           |                                                      2                                     3                         |                  3                                             |
  |           |                                         /       3   \                               x - sin (x)                      |           x - sin (x)                                          |
  \           \                                         \x - sin (x)/                                                                /                                                                /
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
                                                                                                          2                                                                                            
                                                                                             /       3   \                                                                                             
                                                                                             \x - sin (x)/

$$\frac{3 \left(\left(5 x - 3\right) \left(\left(7 \sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \cos{\left(x \right)} + \frac{6 \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right) \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)}}{x - \sin^{3}{\left(x \right)}} - \frac{2 \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right)^{3}}{\left(x - \sin^{3}{\left(x \right)}\right)^{2}}\right) + 15 \left(\sin^{2}{\left(x \right)} - 2 \cos^{2}{\left(x \right)}\right) \sin{\left(x \right)} - \frac{10 \left(3 \sin^{2}{\left(x \right)} \cos{\left(x \right)} - 1\right)^{2}}{x - \sin^{3}{\left(x \right)}}\right)}{\left(x - \sin^{3}{\left(x \right)}\right)^{2}}$$

Simplify

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Derivative of (5x-3)/(sin^3-x)

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The solution