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three /(2sinx-cosx)
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3/(2sinx+cosx)

The derivative of the function/ 3/(2sinx-cosx)

Derivative of 3/(2sinx-cosx)

The solution

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

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

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

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Let $u = 2 \sin{\left(x \right)} - \cos{\left(x \right)}$ .
2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right)$ :
  1. Differentiate $2 \sin{\left(x \right)} - \cos{\left(x \right)}$ term by term:
    1. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      So, the result is: $2 \cos{\left(x \right)}$
    2. The derivative of a constant times a function is the constant times the derivative of the function.
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      So, the result is: $\sin{\left(x \right)}$
    The result is: $\sin{\left(x \right)} + 2 \cos{\left(x \right)}$
  The result of the chain rule is:
  
  $- \frac{\sin{\left(x \right)} + 2 \cos{\left(x \right)}}{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}$
So, the result is: $- \frac{3 \left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)}{\left(2 \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   /                         2\                    
   |    6*(2*cos(x) + sin(x)) |                    
-3*|5 + ----------------------|*(2*cos(x) + sin(x))
   |                        2 |                    
   \    (-cos(x) + 2*sin(x))  /                    
---------------------------------------------------
                                   2               
               (-cos(x) + 2*sin(x))

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

Simplify

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Derivative of 3/(2sinx-cosx)

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The solution