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Derivative of (x+5)/(x+1)
Derivative of -x/2
Derivative of sin(4x)
Derivative of x*2^x
Identical expressions
(2sinxcosx/sin^2x)+2cos^3x/sin^3x
(2 sinus of x co sinus of e of x divide by sinus of squared x) plus 2 co sinus of e of cubed x divide by sinus of cubed x
(2sinxcosx/sin2x)+2cos3x/sin3x
2sinxcosx/sin2x+2cos3x/sin3x
(2sinxcosx/sin²x)+2cos³x/sin³x
(2sinxcosx/sin to the power of 2x)+2cos to the power of 3x/sin to the power of 3x
2sinxcosx/sin^2x+2cos^3x/sin^3x
(2sinxcosx divide by sin^2x)+2cos^3x divide by sin^3x
Similar expressions
(2sinxcosx/sin^2x)-2cos^3x/sin^3x

The derivative of the function/ (2sinxcosx/sin^2x)+2cos^3x/sin^3x

Derivative of (2sinxcosx/sin^2x)+2cos^3x/sin^3x

The solution

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

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

((2*sin(x))*cos(x))/sin(x)^2 + (2*cos(x)^3)/sin(x)^3

Detail solution

Differentiate $\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{2 \cos^{3}{\left(x \right)}}{\sin^{3}{\left(x \right)}}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)}$ and $g{\left(x \right)} = \sin^{2}{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Apply the product rule:
      
      $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
      
      $f{\left(x \right)} = \cos{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      $g{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
      1. The derivative of sine is cosine:
        
        $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
      The result is: $- \sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}$
    So, the result is: $- 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \sin{\left(x \right)}$ .
  2. Apply the power rule: $u^{2}$ goes to $2 u$
  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:
    
    $2 \sin{\left(x \right)} \cos{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{\left(- 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$
2. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = 2 \cos^{3}{\left(x \right)}$ and $g{\left(x \right)} = \sin^{3}{\left(x \right)}$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of a constant times a function is the constant times the derivative of the function.
    1. Let $u = \cos{\left(x \right)}$ .
    2. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \cos{\left(x \right)}$ :
      1. The derivative of cosine is negative sine:
        
        $\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}$
      The result of the chain rule is:
      
      $- 3 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
    So, the result is: $- 6 \sin{\left(x \right)} \cos^{2}{\left(x \right)}$
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  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:
    
    $3 \sin^{2}{\left(x \right)} \cos{\left(x \right)}$
  Now plug in to the quotient rule:
  
  $\frac{- 6 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} - 6 \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{\sin^{6}{\left(x \right)}}$
The result is: $\frac{\left(- 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}\right) \sin^{2}{\left(x \right)} - 4 \sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{- 6 \sin^{4}{\left(x \right)} \cos^{2}{\left(x \right)} - 6 \sin^{2}{\left(x \right)} \cos^{4}{\left(x \right)}}{\sin^{6}{\left(x \right)}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2           2           4           2           2          
- 2*sin (x) + 2*cos (x)   6*cos (x)   4*cos (x)   6*cos (x)*sin(x)
----------------------- - --------- - --------- - ----------------
           2                  4           2              3        
        sin (x)            sin (x)     sin (x)        sin (x)

$$\frac{- 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \sin{\left(x \right)} \cos^{2}{\left(x \right)}}{\sin^{3}{\left(x \right)}} - \frac{4 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} - \frac{6 \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)}}$$

Simplify

The second derivative [src]

  /       2         2           4            2   \       
  |    sin (x) - cos (x)   6*cos (x)   11*cos (x)|       
4*|3 + ----------------- + --------- + ----------|*cos(x)
  |            2               4           2     |       
  \         sin (x)         sin (x)     sin (x)  /       
---------------------------------------------------------
                          sin(x)

$$\frac{4 \left(\frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + 3 + \frac{11 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{6 \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)}}\right) \cos{\left(x \right)}}{\sin{\left(x \right)}}$$

Simplify

The third derivative [src]

   /       2         2            6            2            4           2    /   2         2   \\
   |    sin (x) - cos (x)   30*cos (x)   32*cos (x)   63*cos (x)   3*cos (x)*\sin (x) - cos (x)/|
-4*|3 + ----------------- + ---------- + ---------- + ---------- + -----------------------------|
   |            2               6            2            4                      4              |
   \         sin (x)         sin (x)      sin (x)      sin (x)                sin (x)           /

$$- 4 \left(\frac{\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{3 \left(\sin^{2}{\left(x \right)} - \cos^{2}{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + 3 + \frac{32 \cos^{2}{\left(x \right)}}{\sin^{2}{\left(x \right)}} + \frac{63 \cos^{4}{\left(x \right)}}{\sin^{4}{\left(x \right)}} + \frac{30 \cos^{6}{\left(x \right)}}{\sin^{6}{\left(x \right)}}\right)$$

Simplify

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Derivative of (2sinxcosx/sin^2x)+2cos^3x/sin^3x

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The solution