Derivative of lnx/(x+2)+1

You have entered [src]

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

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

log(x)/(x + 2) + 1

Detail solution

Differentiate $1 + \frac{\log{\left(x \right)}}{x + 2}$ term by term:
1. Apply the quotient rule, which is:
  
  $\frac{d}{d x} \frac{f{\left(x \right)}}{g{\left(x \right)}} = \frac{- f{\left(x \right)} \frac{d}{d x} g{\left(x \right)} + g{\left(x \right)} \frac{d}{d x} f{\left(x \right)}}{g^{2}{\left(x \right)}}$
  
  $f{\left(x \right)} = \log{\left(x \right)}$ and $g{\left(x \right)} = x + 2$ .
  
  To find $\frac{d}{d x} f{\left(x \right)}$ :
  1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  To find $\frac{d}{d x} g{\left(x \right)}$ :
  1. Differentiate $x + 2$ term by term:
    1. The derivative of the constant $2$ is zero.
    2. Apply the power rule: $x$ goes to $1$
    The result is: $1$
  Now plug in to the quotient rule:
  
  $\frac{- \log{\left(x \right)} + \frac{x + 2}{x}}{\left(x + 2\right)^{2}}$
2. The derivative of the constant $1$ is zero.
The result is: $\frac{- \log{\left(x \right)} + \frac{x + 2}{x}}{\left(x + 2\right)^{2}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

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

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

Simplify

Other calculators

How to use it?

Identical expressions

Similar expressions

Derivative of lnx/(x+2)+1

The graph:

Piecewise:

The solution