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The derivative of the function/ lnx/2√x+√x/x

Derivative of lnx/2√x+√x/x

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
                 ___
log(x)   ___   \/ x 
------*\/ x  + -----
  2              x
xlog⁡(x)2+xx\sqrt{x} \frac{\log{\left(x \right)}}{2} + \frac{\sqrt{x}}{x} 
(log(x)/2)*sqrt(x) + sqrt(x)/x
Detail solution

  1. Differentiate xlog⁡(x)2+xx\sqrt{x} \frac{\log{\left(x \right)}}{2} + \frac{\sqrt{x}}{x} term by term:

    1. 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)=xlog⁡(x)f{\left(x \right)} = \sqrt{x} \log{\left(x \right)} and g(x)=2g{\left(x \right)} = 2.

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

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

        1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

        The result is: log⁡(x)2x+1x\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}

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

      1. The derivative of the constant 22 is zero.

      Now plug in to the quotient rule:

      log⁡(x)4x+12x\frac{\log{\left(x \right)}}{4 \sqrt{x}} + \frac{1}{2 \sqrt{x}}

    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)=xf{\left(x \right)} = \sqrt{x} and g(x)=xg{\left(x \right)} = 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}}

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

      1. Apply the power rule: xx goes to 11

      Now plug in to the quotient rule:

      −12x32- \frac{1}{2 x^{\frac{3}{2}}}

    The result is: log⁡(x)4x+12x−12x32\frac{\log{\left(x \right)}}{4 \sqrt{x}} + \frac{1}{2 \sqrt{x}} - \frac{1}{2 x^{\frac{3}{2}}}

  2. Now simplify:

    x(log⁡(x)+2)−24x32\frac{x \left(\log{\left(x \right)} + 2\right) - 2}{4 x^{\frac{3}{2}}}

The answer is:

x(log⁡(x)+2)−24x32\frac{x \left(\log{\left(x \right)} + 2\right) - 2}{4 x^{\frac{3}{2}}}

The graph
02468-8-6-4-2-1010-2020
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
   1        1       log(x)
------- - ------ + -------
    ___      3/2       ___
2*\/ x    2*x      4*\/ x
log⁡(x)4x+12x−12x32\frac{\log{\left(x \right)}}{4 \sqrt{x}} + \frac{1}{2 \sqrt{x}} - \frac{1}{2 x^{\frac{3}{2}}}
Simplify
The second derivative [src] 
          6
-log(x) + -
          x
-----------
      3/2  
   8*x
−log⁡(x)+6x8x32\frac{- \log{\left(x \right)} + \frac{6}{x}}{8 x^{\frac{3}{2}}}
Simplify
The third derivative [src] 
     30           
-2 - -- + 3*log(x)
     x            
------------------
         5/2      
     16*x
3log⁡(x)−2−30x16x52\frac{3 \log{\left(x \right)} - 2 - \frac{30}{x}}{16 x^{\frac{5}{2}}}
Simplify
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