Mister Exam

Derivative of x*ln(x+1)

Function f() - derivative -N order at the point
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
x*log(x + 1)
$$x \log{\left(x + 1 \right)}$$
d               
--(x*log(x + 1))
dx              
$$\frac{d}{d x} x \log{\left(x + 1 \right)}$$
Detail solution
  1. Apply the product rule:

    ; to find :

    1. Apply the power rule: goes to

    ; to find :

    1. Let .

    2. The derivative of is .

    3. Then, apply the chain rule. Multiply by :

      1. Differentiate term by term:

        1. Apply the power rule: goes to

        2. The derivative of the constant is zero.

        The result is:

      The result of the chain rule is:

    The result is:

  2. Now simplify:


The answer is:

The graph
The first derivative [src]
  x               
----- + log(x + 1)
x + 1             
$$\frac{x}{x + 1} + \log{\left(x + 1 \right)}$$
The second derivative [src]
      x  
2 - -----
    1 + x
---------
  1 + x  
$$\frac{- \frac{x}{x + 1} + 2}{x + 1}$$
The third derivative [src]
      2*x 
-3 + -----
     1 + x
----------
        2 
 (1 + x)  
$$\frac{\frac{2 x}{x + 1} - 3}{\left(x + 1\right)^{2}}$$
The graph
Derivative of x*ln(x+1)