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Derivative of sqrt(x)*log(x,2)

You have entered [src]

  ___ log(x)
\/ x *------
      log(2)

\sqrt{x} \frac{\log{\left(x \right)}}{\log{\left(2 \right)}}

sqrt(x)*(log(x)/log(2))

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = \sqrt{x}$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $\sqrt{x}$ goes to $\frac{1}{2 \sqrt{x}}$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. The derivative of the constant $\log{\left(2 \right)}$ is zero.
Now plug in to the quotient rule:

$\frac{\frac{\log{\left(x \right)}}{2 \sqrt{x}} + \frac{1}{\sqrt{x}}}{\log{\left(2 \right)}}$
Now simplify:

$\frac{\log{\left(x \right)} + 2}{2 \sqrt{x} \log{\left(2 \right)}}$

The answer is:

\frac{\log{\left(x \right)} + 2}{2 \sqrt{x} \log{\left(2 \right)}}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

   -log(x)   
-------------
   3/2       
4*x   *log(2)

- \frac{\log{\left(x \right)}}{4 x^{\frac{3}{2}} \log{\left(2 \right)}}

Simplify

The third derivative [src]

-2 + 3*log(x)
-------------
   5/2       
8*x   *log(2)

\frac{3 \log{\left(x \right)} - 2}{8 x^{\frac{5}{2}} \log{\left(2 \right)}}

Simplify