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The derivative of the function/ sqrtx*tg(x)

Derivative of sqrtx*tg(x)

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

You have entered [src] 
  ___       
\/ x *tan(x)
xtan⁡(x)\sqrt{x} \tan{\left(x \right)} 
sqrt(x)*tan(x)
Detail solution

  1. 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)=tan⁡(x)g{\left(x \right)} = \tan{\left(x \right)}; to find ddxg(x)\frac{d}{d x} g{\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 is: x(sin⁡2(x)+cos⁡2(x))cos⁡2(x)+tan⁡(x)2x\frac{\sqrt{x} \left(\sin^{2}{\left(x \right)} + \cos^{2}{\left(x \right)}\right)}{\cos^{2}{\left(x \right)}} + \frac{\tan{\left(x \right)}}{2 \sqrt{x}}

  2. Now simplify:

    x+sin⁡(2x)4xcos⁡2(x)\frac{x + \frac{\sin{\left(2 x \right)}}{4}}{\sqrt{x} \cos^{2}{\left(x \right)}}

The answer is:

x+sin⁡(2x)4xcos⁡2(x)\frac{x + \frac{\sin{\left(2 x \right)}}{4}}{\sqrt{x} \cos^{2}{\left(x \right)}}

The graph
02468-8-6-4-2-1010-20002000
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
  ___ /       2   \    tan(x)
\/ x *\1 + tan (x)/ + -------
                          ___
                      2*\/ x
x(tan⁡2(x)+1)+tan⁡(x)2x\sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right) + \frac{\tan{\left(x \right)}}{2 \sqrt{x}}
Simplify
The second derivative [src] 
       2                                           
1 + tan (x)   tan(x)       ___ /       2   \       
----------- - ------ + 2*\/ x *\1 + tan (x)/*tan(x)
     ___         3/2                               
   \/ x       4*x
2x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1x−tan⁡(x)4x322 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x}} - \frac{\tan{\left(x \right)}}{4 x^{\frac{3}{2}}}
Simplify
The third derivative [src] 
    /       2   \                                                        /       2   \       
  3*\1 + tan (x)/   3*tan(x)       ___ /       2   \ /         2   \   3*\1 + tan (x)/*tan(x)
- --------------- + -------- + 2*\/ x *\1 + tan (x)/*\1 + 3*tan (x)/ + ----------------------
          3/2           5/2                                                      ___         
       4*x           8*x                                                       \/ x
2x(tan⁡2(x)+1)(3tan⁡2(x)+1)+3(tan⁡2(x)+1)tan⁡(x)x−3(tan⁡2(x)+1)4x32+3tan⁡(x)8x522 \sqrt{x} \left(\tan^{2}{\left(x \right)} + 1\right) \left(3 \tan^{2}{\left(x \right)} + 1\right) + \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}}{\sqrt{x}} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{4 x^{\frac{3}{2}}} + \frac{3 \tan{\left(x \right)}}{8 x^{\frac{5}{2}}}
Simplify
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