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The derivative of the function/ 0.1x^2-x*ln(x)

Derivative of 0.1x^2-x*ln(x)

The solution

You have entered [src]

 2           
x            
-- - x*log(x)
10

$$\frac{x^{2}}{10} - x \log{\left(x \right)}$$

  / 2           \
d |x            |
--|-- - x*log(x)|
dx\10           /

$$\frac{d}{d x} \left(\frac{x^{2}}{10} - x \log{\left(x \right)}\right)$$

Transform

Detail solution

Differentiate $\frac{x^{2}}{10} - x \log{\left(x \right)}$ term by term:
1. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Apply the power rule: $x^{2}$ goes to $2 x$
  So, the result is: $\frac{x}{5}$
2. 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: $- \log{\left(x \right)} - 1$
The result is: $\frac{x}{5} - \log{\left(x \right)} - 1$

The answer is:

$\frac{x}{5} - \log{\left(x \right)} - 1$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              x
-1 - log(x) + -
              5

$$\frac{x}{5} - \log{\left(x \right)} - 1$$

Simplify

The second derivative [src]

1   1
- - -
5   x

$$\frac{1}{5} - \frac{1}{x}$$

Simplify

The third derivative [src]

1 
--
 2
x

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

Simplify

The graph

Derivative of 0.1x^2-x*ln(x)