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Derivative of (x+1)(ln(x+1))^2

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           2       
(x + 1)*log (x + 1)

$$\left(x + 1\right) \log{\left(x + 1 \right)}^{2}$$

(x + 1)*log(x + 1)^2

Detail solution

Apply the product rule:

$\frac{d}{d x} f{\left(x \right)} g{\left(x \right)} = f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}$

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

$\left(\log{\left(x + 1 \right)} + 2\right) \log{\left(x + 1 \right)}$

The answer is:

$\left(\log{\left(x + 1 \right)} + 2\right) \log{\left(x + 1 \right)}$

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

   2                      
log (x + 1) + 2*log(x + 1)

$$\log{\left(x + 1 \right)}^{2} + 2 \log{\left(x + 1 \right)}$$

Simplify

The second derivative [src]

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

$$\frac{2 \left(\log{\left(x + 1 \right)} + 1\right)}{x + 1}$$

Simplify

The third derivative [src]

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

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

Simplify