Mister Exam

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The derivative of the function/ x*loge(x)

Derivative of x*loge(x)

The solution

You have entered [src]

x*log(x)
--------
   / 1\ 
log\e /

$$\frac{x \log{\left(x \right)}}{\log{\left(e^{1} \right)}}$$

(x*log(x))/log(exp(1))

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

1 + log(x)
----------
    / 1\  
 log\e /

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

Simplify

The second derivative [src]

    1    
---------
     / 1\
x*log\e /

$$\frac{1}{x \log{\left(e^{1} \right)}}$$

Simplify

The third derivative [src]

   -1     
----------
 2    / 1\
x *log\e /

$$- \frac{1}{x^{2} \log{\left(e^{1} \right)}}$$

Simplify