Derivative of ln(1+x)/x

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

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

d /log(1 + x)\
--|----------|
dx\    x     /

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

Transform

Detail solution

Apply the quotient rule, which is:

$\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$

$f{\left(x \right)} = \log{\left(x + 1 \right)}$ and $g{\left(x \right)} = x$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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. The derivative of the constant $1$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  The result of the chain rule is:
  
  $\frac{1}{x + 1}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x$ goes to $1$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

   2       6*log(1 + x)       3            6     
-------- - ------------ + ---------- + ----------
       3         3                 2    2        
(1 + x)         x         x*(1 + x)    x *(1 + x)
-------------------------------------------------
                        x

$$\frac{- \frac{6 \log{\left(x + 1 \right)}}{x^{3}} + \frac{2}{\left(x + 1\right)^{3}} + \frac{3}{x \left(x + 1\right)^{2}} + \frac{6}{x^{2} \left(x + 1\right)}}{x}$$

Simplify

The graph

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Derivative of ln(1+x)/x

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Piecewise:

The solution