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Identical expressions
two ^tgx+(sqrt(x))*sin2x
2 to the power of tgx plus ( square root of (x)) multiply by sinus of 2x
two to the power of tgx plus ( square root of (x)) multiply by sinus of 2x
2^tgx+(√(x))*sin2x
2tgx+(sqrt(x))*sin2x
2tgx+sqrtx*sin2x
2^tgx+(sqrt(x))sin2x
2tgx+(sqrt(x))sin2x
2tgx+sqrtxsin2x
2^tgx+sqrtxsin2x
Similar expressions
2^tgx-(sqrt(x))*sin2x

The derivative of the function/ 2^tgx+(sqrt(x))*sin2x

Derivative of 2^tgx+(sqrt(x))*sin2x

The solution

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 tan(x)     ___         
2       + \/ x *sin(2*x)

$$2^{\tan{\left(x \right)}} + \sqrt{x} \sin{\left(2 x \right)}$$

2^tan(x) + sqrt(x)*sin(2*x)

Detail solution

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

sin(2*x)       ___             tan(x) /       2   \       
-------- + 2*\/ x *cos(2*x) + 2      *\1 + tan (x)/*log(2)
    ___                                                   
2*\/ x

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

Simplify

The second derivative [src]

                                                                  2                                                
      ___            2*cos(2*x)   sin(2*x)    tan(x) /       2   \     2         tan(x) /       2   \              
- 4*\/ x *sin(2*x) + ---------- - -------- + 2      *\1 + tan (x)/ *log (2) + 2*2      *\1 + tan (x)/*log(2)*tan(x)
                         ___          3/2                                                                          
                       \/ x        4*x

$$2^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} + 2 \cdot 2^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan{\left(x \right)} - 4 \sqrt{x} \sin{\left(2 x \right)} + \frac{2 \cos{\left(2 x \right)}}{\sqrt{x}} - \frac{\sin{\left(2 x \right)}}{4 x^{\frac{3}{2}}}$$

Simplify

The third derivative [src]

                                                                                 3                                  2                                                                          2               
      ___            6*sin(2*x)   3*cos(2*x)   3*sin(2*x)    tan(x) /       2   \     3         tan(x) /       2   \              tan(x)    2    /       2   \             tan(x) /       2   \     2          
- 8*\/ x *cos(2*x) - ---------- - ---------- + ---------- + 2      *\1 + tan (x)/ *log (2) + 2*2      *\1 + tan (x)/ *log(2) + 4*2      *tan (x)*\1 + tan (x)/*log(2) + 6*2      *\1 + tan (x)/ *log (2)*tan(x)
                         ___           3/2          5/2                                                                                                                                                        
                       \/ x         2*x          8*x

$$2^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right)^{3} \log{\left(2 \right)}^{3} + 6 \cdot 2^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)}^{2} \tan{\left(x \right)} + 2 \cdot 2^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right)^{2} \log{\left(2 \right)} + 4 \cdot 2^{\tan{\left(x \right)}} \left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(2 \right)} \tan^{2}{\left(x \right)} - 8 \sqrt{x} \cos{\left(2 x \right)} - \frac{6 \sin{\left(2 x \right)}}{\sqrt{x}} - \frac{3 \cos{\left(2 x \right)}}{2 x^{\frac{3}{2}}} + \frac{3 \sin{\left(2 x \right)}}{8 x^{\frac{5}{2}}}$$

Simplify

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Derivative of 2^tgx+(sqrt(x))*sin2x

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