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Derivative of:
Derivative of sqrt(cos(x))
Derivative of sqrt(x-1)/(sqrt(x^2-x)-1)
Derivative of 5*tan(x)
Derivative of (4x+1)/(3x-1)
Identical expressions
lnx/ two *x^ zero . five
lnx divide by 2 multiply by x to the power of 0.5
lnx divide by two multiply by x to the power of zero . five
lnx/2*x0.5
lnx/2x^0.5
lnx/2x0.5
lnx divide by 2*x^0.5

The derivative of the function/ lnx/2*x^0.5

Derivative of lnx/2*x^0.5

The solution

You have entered [src]

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

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

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

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 $g{\left(x \right)} = 2$ .

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 $2$ is zero.
Now plug in to the quotient rule:

$\frac{\log{\left(x \right)}}{4 \sqrt{x}} + \frac{1}{2 \sqrt{x}}$
Now simplify:

$\frac{\log{\left(x \right)} + 2}{4 \sqrt{x}}$

The answer is:

$\frac{\log{\left(x \right)} + 2}{4 \sqrt{x}}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   1       log(x)
------- + -------
    ___       ___
2*\/ x    4*\/ x

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

$$\frac{3 \log{\left(x \right)} - 2}{16 x^{\frac{5}{2}}}$$

Simplify