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The derivative of the function/ (sin(x)+x^2)/cot(2*x)

Derivative of (sin(x)+x^2)/cot(2*x)

The solution

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          2
sin(x) + x 
-----------
  cot(2*x)

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

(sin(x) + x^2)/cot(2*x)

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = x^{2} + \sin{\left(x \right)}$ and $g{\left(x \right)} = \cot{\left(2 x \right)}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $x^{2} + \sin{\left(x \right)}$ term by term:
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  2. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  The result is: $2 x + \cos{\left(x \right)}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. There are multiple ways to do this derivative.
  Method #1
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(2 x \right)} = \frac{1}{\tan{\left(2 x \right)}}$
  2. Let $u = \tan{\left(2 x \right)}$ .
  3. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  4. Then, apply the chain rule. Multiply by $\frac{d}{d x} \tan{\left(2 x \right)}$ :
    1. Rewrite the function to be differentiated:
      
      $\tan{\left(2 x \right)} = \frac{\sin{\left(2 x \right)}}{\cos{\left(2 x \right)}}$
    2. Apply the quotient rule, which is:
      
      $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
      
      $f{\left(x \right)} = \sin{\left(2 x \right)}$ and $g{\left(x \right)} = \cos{\left(2 x \right)}$ .
      
      To find $\frac{d}{d x} f{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
      
      To find $\frac{d}{d x} g{\left(x \right)}$ :
      
      Let $u = 2 x$ .
      
      The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
      
      Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $- 2 \sin{\left(2 x \right)}$
      
      Now plug in to the quotient rule:
      
      $\frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}}$
    The result of the chain rule is:
    
    $- \frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}}$
  Method #2
  1. Rewrite the function to be differentiated:
    
    $\cot{\left(2 x \right)} = \frac{\cos{\left(2 x \right)}}{\sin{\left(2 x \right)}}$
  2. Apply the quotient rule, which is:
    
    $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
    
    $f{\left(x \right)} = \cos{\left(2 x \right)}$ and $g{\left(x \right)} = \sin{\left(2 x \right)}$ .
    
    To find $\frac{d}{d x} f{\left(x \right)}$ :
    1. Let $u = 2 x$ .
    2. The derivative of cosine is negative sine:
      
      $\frac{d}{d u} \cos{\left(u \right)} = - \sin{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $- 2 \sin{\left(2 x \right)}$
    To find $\frac{d}{d x} g{\left(x \right)}$ :
    1. Let $u = 2 x$ .
    2. The derivative of sine is cosine:
      
      $\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} 2 x$ :
      
      The derivative of a constant times a function is the constant times the derivative of the function.
      
      Apply the power rule: $x$ goes to $1$
      
      So, the result is: $2$
      
      The result of the chain rule is:
      
      $2 \cos{\left(2 x \right)}$
    Now plug in to the quotient rule:
    
    $\frac{- 2 \sin^{2}{\left(2 x \right)} - 2 \cos^{2}{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)}}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

               /         2     \ /          2\
2*x + cos(x)   \2 + 2*cot (2*x)/*\sin(x) + x /
------------ + -------------------------------
  cot(2*x)                   2                
                          cot (2*x)

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

Simplify

The second derivative [src]

               /       2     \                                    /            2     \              
             4*\1 + cot (2*x)/*(2*x + cos(x))     /       2     \ |     1 + cot (2*x)| / 2         \
2 - sin(x) + -------------------------------- + 8*\1 + cot (2*x)/*|-1 + -------------|*\x  + sin(x)/
                         cot(2*x)                                 |          2       |              
                                                                  \       cot (2*x)  /              
----------------------------------------------------------------------------------------------------
                                              cot(2*x)

$$\frac{\frac{4 \left(2 x + \cos{\left(x \right)}\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}{\cot{\left(2 x \right)}} + 8 \left(x^{2} + \sin{\left(x \right)}\right) \left(\frac{\cot^{2}{\left(2 x \right)} + 1}{\cot^{2}{\left(2 x \right)}} - 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right) - \sin{\left(x \right)} + 2}{\cot{\left(2 x \right)}}$$

Simplify

The third derivative [src]

                                                                                                                                                 /            2     \               
                                                                                                                                 /       2     \ |     1 + cot (2*x)|               
                              /                                   2                    3\                                     24*\1 + cot (2*x)/*|-1 + -------------|*(2*x + cos(x))
                              |                    /       2     \      /       2     \ |     /       2     \                                    |          2       |               
   cos(x)       / 2         \ |         2        5*\1 + cot (2*x)/    3*\1 + cot (2*x)/ |   6*\1 + cot (2*x)/*(-2 + sin(x))                      \       cot (2*x)  /               
- -------- + 16*\x  + sin(x)/*|2 + 2*cot (2*x) - ------------------ + ------------------| - ------------------------------- + ------------------------------------------------------
  cot(2*x)                    |                         2                    4          |                 2                                          cot(2*x)                       
                              \                      cot (2*x)            cot (2*x)     /              cot (2*x)

$$\frac{24 \left(2 x + \cos{\left(x \right)}\right) \left(\frac{\cot^{2}{\left(2 x \right)} + 1}{\cot^{2}{\left(2 x \right)}} - 1\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}{\cot{\left(2 x \right)}} + 16 \left(x^{2} + \sin{\left(x \right)}\right) \left(\frac{3 \left(\cot^{2}{\left(2 x \right)} + 1\right)^{3}}{\cot^{4}{\left(2 x \right)}} - \frac{5 \left(\cot^{2}{\left(2 x \right)} + 1\right)^{2}}{\cot^{2}{\left(2 x \right)}} + 2 \cot^{2}{\left(2 x \right)} + 2\right) - \frac{6 \left(\sin{\left(x \right)} - 2\right) \left(\cot^{2}{\left(2 x \right)} + 1\right)}{\cot^{2}{\left(2 x \right)}} - \frac{\cos{\left(x \right)}}{\cot{\left(2 x \right)}}$$

Simplify

The graph

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Derivative of (sin(x)+x^2)/cot(2*x)

The graph:

Piecewise:

The solution

Method #1

Method #2