Mister Exam

Lang:

The derivative of the function/ x*ln(x+1)

Derivative of x*ln(x+1)

The solution

You have entered [src]

x*log(x + 1)

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

d               
--(x*log(x + 1))
dx

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

Transform

Detail solution

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 $1$
$g{\left(x \right)} = \log{\left(x + 1 \right)}$ ; to find $\frac{d}{d x} g{\left(x \right)}$ :
1. Let $u = x + 1$ .
2. The derivative of $\log{\left(u \right)}$ is $\frac{1}{u}$ .
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(x + 1\right)$ :
  1. Differentiate $x + 1$ term by term:
    1. Apply the power rule: $x$ goes to $1$
    2. The derivative of the constant $1$ is zero.
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{x + 1}$
The result is: $\frac{x}{x + 1} + \log{\left(x + 1 \right)}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

      x  
2 - -----
    1 + x
---------
  1 + x

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

Simplify

The third derivative [src]

      2*x 
-3 + -----
     1 + x
----------
        2 
 (1 + x)

$$\frac{\frac{2 x}{x + 1} - 3}{\left(x + 1\right)^{2}}$$

Simplify

The graph

Derivative of x*ln(x+1)