Derivative of exp(x*ln(x+1))

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 x*log(x + 1)
e

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

exp(x*log(x + 1))

Detail solution

Let $u = x \log{\left(x + 1 \right)}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} x \log{\left(x + 1 \right)}$ :
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 $1$
  $g{\left(x \right)} = \log{\left(x + 1 \right)}$ ; 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: $\frac{x}{x + 1} + \log{\left(x + 1 \right)}$
The result of the chain rule is:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

/                               x  \              
|                    2   -2 + -----|              
|/  x               \         1 + x|  x*log(1 + x)
||----- + log(1 + x)|  - ----------|*e            
\\1 + x             /      1 + x   /

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

Simplify

The third derivative [src]

/                              2*x      /       x  \ /  x               \\              
|                    3   -3 + -----   3*|-2 + -----|*|----- + log(1 + x)||              
|/  x               \         1 + x     \     1 + x/ \1 + x             /|  x*log(1 + x)
||----- + log(1 + x)|  + ---------- - -----------------------------------|*e            
|\1 + x             /            2                   1 + x               |              
\                         (1 + x)                                        /

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

Simplify

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Derivative of exp(x*ln(x+1))

The graph:

Piecewise:

The solution