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The derivative of the function/ (xlnx)/2

Derivative of (xlnx)/2

The solution

You have entered [src]

x*log(x)
--------
   2

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

(x*log(x))/2

Detail solution

The derivative of a constant times a function is the constant times the derivative of the function.
1. Apply the product rule:
  
  $\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$
  
  $f{\left(x \right)} = x$ ; to find $\frac{d}{d x} f{\left(x \right)}$ :
  1. Apply the power rule: $x$ goes to $1$
  $g{\left(x \right)} = \log{\left(x \right)}$ ; 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: $\log{\left(x \right)} + 1$
So, the result is: $\frac{\log{\left(x \right)}}{2} + \frac{1}{2}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1   log(x)
- + ------
2     2

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

Simplify

The second derivative [src]

 1 
---
2*x

$$\frac{1}{2 x}$$

Simplify

The third derivative [src]

-1  
----
   2
2*x

$$- \frac{1}{2 x^{2}}$$

Simplify