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sinx*e^cosx-1/tg2x

The derivative of the function/ sinx*e^cosx+1/tg2x

Derivative of sinx*e^cosx+1/tg2x

The solution

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        cos(x)      1    
sin(x)*E       + --------
                 tan(2*x)

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

Transform

Detail solution

Differentiate $e^{\cos{\left(x \right)}} \sin{\left(x \right)} + \frac{1}{\tan{\left(2 x \right)}}$ term by term:
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sin{\left(x \right)}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of sine is cosine:
    
    $\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}$
  $g{\left(x \right)} = e^{\cos{\left(x \right)}}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Let $u = \cos{\left(x \right)}$ .
  2. The derivative of $e^{u}$ is itself.
  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:
    
    $- e^{\cos{\left(x \right)}} \sin{\left(x \right)}$
  The result is: $- e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} + e^{\cos{\left(x \right)}} \cos{\left(x \right)}$
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 $\frac{d}{d x} f{\left(x \right)}$ :
    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 $\frac{d}{d x} 2 x$ :
      1. 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 $\frac{d}{d x} g{\left(x \right)}$ :
    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 $\frac{d}{d x} 2 x$ :
      1. 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:
  
  $- \frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}}$
The result is: $- \frac{2 \sin^{2}{\left(2 x \right)} + 2 \cos^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)} \tan^{2}{\left(2 x \right)}} - e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} + e^{\cos{\left(x \right)}} \cos{\left(x \right)}$
Now simplify:

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

The answer is:

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

The graph

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Plot the graph f'(x)

The first derivative [src]

          2                                        
-2 - 2*tan (2*x)           cos(x)      2     cos(x)
---------------- + cos(x)*e       - sin (x)*e      
      2                                            
   tan (2*x)

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

Simplify

The second derivative [src]

                                                                        2                          
                                     /       2     \     /       2     \                           
   3     cos(x)    cos(x)          8*\1 + tan (2*x)/   8*\1 + tan (2*x)/              cos(x)       
sin (x)*e       - e      *sin(x) - ----------------- + ------------------ - 3*cos(x)*e      *sin(x)
                                        tan(2*x)              3                                    
                                                           tan (2*x)

$$\frac{8 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}}{\tan^{3}{\left(2 x \right)}} - \frac{8 \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan{\left(2 x \right)}} + e^{\cos{\left(x \right)}} \sin^{3}{\left(x \right)} - 3 e^{\cos{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)} - e^{\cos{\left(x \right)}} \sin{\left(x \right)}$$

Simplify

The third derivative [src]

                                                                          3                                                             2                           
                                                           /       2     \                                               /       2     \                            
            2           4     cos(x)           cos(x)   48*\1 + tan (2*x)/         2     cos(x)        2     cos(x)   80*\1 + tan (2*x)/         2            cos(x)
-32 - 32*tan (2*x) - sin (x)*e       - cos(x)*e       - ------------------- - 3*cos (x)*e       + 4*sin (x)*e       + ------------------- + 6*sin (x)*cos(x)*e      
                                                                4                                                             2                                     
                                                             tan (2*x)                                                     tan (2*x)

$$- \frac{48 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{3}}{\tan^{4}{\left(2 x \right)}} + \frac{80 \left(\tan^{2}{\left(2 x \right)} + 1\right)^{2}}{\tan^{2}{\left(2 x \right)}} - e^{\cos{\left(x \right)}} \sin^{4}{\left(x \right)} + 6 e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} \cos{\left(x \right)} + 4 e^{\cos{\left(x \right)}} \sin^{2}{\left(x \right)} - 3 e^{\cos{\left(x \right)}} \cos^{2}{\left(x \right)} - e^{\cos{\left(x \right)}} \cos{\left(x \right)} - 32 \tan^{2}{\left(2 x \right)} - 32$$

Simplify

The graph

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Derivative of sinx*e^cosx+1/tg2x

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The solution