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

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

Function f() - derivative -N order at the point
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The solution

You have entered [src] 
 tan(x)     ___         
2       + \/ x *sin(2*x)
2tan⁡(x)+xsin⁡(2x)2^{\tan{\left(x \right)}} + \sqrt{x} \sin{\left(2 x \right)} 
2^tan(x) + sqrt(x)*sin(2*x)
Detail solution

  1. Differentiate 2tan⁡(x)+xsin⁡(2x)2^{\tan{\left(x \right)}} + \sqrt{x} \sin{\left(2 x \right)} term by term:

    1. Let u=tan⁡(x)u = \tan{\left(x \right)}.

    2. ddu2u=2ulog⁡(2)\frac{d}{d u} 2^{u} = 2^{u} \log{\left(2 \right)}

    3. Then, apply the chain rule. Multiply by ddxtan⁡(x)\frac{d}{d x} \tan{\left(x \right)}:

      1. Rewrite the function to be differentiated:

        tan⁡(x)=sin⁡(x)cos⁡(x)\tan{\left(x \right)} = \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}}

      2. Apply the quotient rule, which is:

        ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=sin⁡(x)f{\left(x \right)} = \sin{\left(x \right)} and g(x)=cos⁡(x)g{\left(x \right)} = \cos{\left(x \right)}.

        To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

        1. The derivative of sine is cosine:

          ddxsin⁡(x)=cos⁡(x)\frac{d}{d x} \sin{\left(x \right)} = \cos{\left(x \right)}

        To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

        1. The derivative of cosine is negative sine:

          ddxcos⁡(x)=−sin⁡(x)\frac{d}{d x} \cos{\left(x \right)} = - \sin{\left(x \right)}

        Now plug in to the quotient rule:

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

      The result of the chain rule is:

      2tan⁡(x)(sin⁡2(x)+cos⁡2(x))log⁡(2)cos⁡2(x)\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:

      ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = \sqrt{x}; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

      1. Apply the power rule: x\sqrt{x} goes to 12x\frac{1}{2 \sqrt{x}}

      g(x)=sin⁡(2x)g{\left(x \right)} = \sin{\left(2 x \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

      1. Let u=2xu = 2 x.

      2. The derivative of sine is cosine:

        ddusin⁡(u)=cos⁡(u)\frac{d}{d u} \sin{\left(u \right)} = \cos{\left(u \right)}

      3. Then, apply the chain rule. Multiply by ddx2x\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: xx goes to 11

          So, the result is: 22

        The result of the chain rule is:

        2cos⁡(2x)2 \cos{\left(2 x \right)}

      The result is: 2xcos⁡(2x)+sin⁡(2x)2x2 \sqrt{x} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{2 \sqrt{x}}

    The result is: 2tan⁡(x)(sin⁡2(x)+cos⁡2(x))log⁡(2)cos⁡2(x)+2xcos⁡(2x)+sin⁡(2x)2x\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}}

  2. Now simplify:

    2tan⁡(x)log⁡(2)cos⁡2(x)+2xcos⁡(2x)+sin⁡(x)cos⁡(x)x\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:

2tan⁡(x)log⁡(2)cos⁡2(x)+2xcos⁡(2x)+sin⁡(x)cos⁡(x)x\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
02468-8-6-4-2-1010-250000000000250000000000
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
2tan⁡(x)(tan⁡2(x)+1)log⁡(2)+2xcos⁡(2x)+sin⁡(2x)2x2^{\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
2tan⁡(x)(tan⁡2(x)+1)2log⁡(2)2+2⋅2tan⁡(x)(tan⁡2(x)+1)log⁡(2)tan⁡(x)−4xsin⁡(2x)+2cos⁡(2x)x−sin⁡(2x)4x322^{\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
2tan⁡(x)(tan⁡2(x)+1)3log⁡(2)3+6⋅2tan⁡(x)(tan⁡2(x)+1)2log⁡(2)2tan⁡(x)+2⋅2tan⁡(x)(tan⁡2(x)+1)2log⁡(2)+4⋅2tan⁡(x)(tan⁡2(x)+1)log⁡(2)tan⁡2(x)−8xcos⁡(2x)−6sin⁡(2x)x−3cos⁡(2x)2x32+3sin⁡(2x)8x522^{\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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