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Identical expressions
y=sinx/(six -2x)
y equally sinus of x divide by (6 minus 2x)
y equally sinus of x divide by (six minus 2x)
y=sinx/6-2x
y=sinx divide by (6-2x)
Similar expressions
y=sinx/(6+2x)

The derivative of the function/ y=sinx/(6-2x)

Derivative of y=sinx/(6-2x)

The solution

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 sin(x)
-------
6 - 2*x

$$\frac{\sin{\left(x \right)}}{6 - 2 x}$$

sin(x)/(6 - 2*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \sin{\left(x \right)}$ and $g{\left(x \right)} = 6 - 2 x$ .

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 $6 - 2 x$ term by term:
  1. The derivative of the constant $6$ 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: $-2$
  The result is: $-2$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

 cos(x)    2*sin(x) 
------- + ----------
6 - 2*x            2
          (6 - 2*x)

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

Simplify

The second derivative [src]

sin(x)   cos(x)     sin(x) 
------ + ------ - ---------
  2      -3 + x           2
                  (-3 + x) 
---------------------------
           -3 + x

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

Simplify

The third derivative [src]

cos(x)    3*cos(x)    3*sin(x)    3*sin(x) 
------ - --------- + --------- - ----------
  2              2           3   2*(-3 + x)
         (-3 + x)    (-3 + x)              
-------------------------------------------
                   -3 + x

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

Simplify

The graph

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Derivative of y=sinx/(6-2x)

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