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)
xlog(x+1)x \log{\left(x + 1 \right)}
d               
--(x*log(x + 1))
dx              
ddxxlog(x+1)\frac{d}{d x} x \log{\left(x + 1 \right)}
Detail solution
  1. Apply the product rule:

    ddxf(x)g(x)=f(x)ddxg(x)+g(x)ddxf(x)\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(x)=xf{\left(x \right)} = x; to find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. Apply the power rule: xx goes to 11

    g(x)=log(x+1)g{\left(x \right)} = \log{\left(x + 1 \right)}; to find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Let u=x+1u = x + 1.

    2. The derivative of log(u)\log{\left(u \right)} is 1u\frac{1}{u}.

    3. Then, apply the chain rule. Multiply by ddx(x+1)\frac{d}{d x} \left(x + 1\right):

      1. Differentiate x+1x + 1 term by term:

        1. Apply the power rule: xx goes to 11

        2. The derivative of the constant 11 is zero.

        The result is: 11

      The result of the chain rule is:

      1x+1\frac{1}{x + 1}

    The result is: xx+1+log(x+1)\frac{x}{x + 1} + \log{\left(x + 1 \right)}

  2. Now simplify:

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


The answer is:

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

The graph
02468-8-6-4-2-1010-5050
The first derivative [src]
  x               
----- + log(x + 1)
x + 1             
xx+1+log(x+1)\frac{x}{x + 1} + \log{\left(x + 1 \right)}
The second derivative [src]
      x  
2 - -----
    1 + x
---------
  1 + x  
xx+1+2x+1\frac{- \frac{x}{x + 1} + 2}{x + 1}
The third derivative [src]
      2*x 
-3 + -----
     1 + x
----------
        2 
 (1 + x)  
2xx+13(x+1)2\frac{\frac{2 x}{x + 1} - 3}{\left(x + 1\right)^{2}}
The graph
Derivative of x*ln(x+1)