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

Derivative of ln(x)/(x+1)

Function f() - derivative -N order at the point
v

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The solution

You have entered [src] 
log(x)
------
x + 1
log⁡(x)x+1\frac{\log{\left(x \right)}}{x + 1} 
d /log(x)\
--|------|
dx\x + 1 /
ddxlog⁡(x)x+1\frac{d}{d x} \frac{\log{\left(x \right)}}{x + 1}
Transform
Detail solution

  1. Apply the quotient rule, which is:

    ddxf(x)g(x)=−f(x)ddxg(x)+g(x)ddxf(x)g2(x)\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(x)=log⁡(x)f{\left(x \right)} = \log{\left(x \right)} and g(x)=x+1g{\left(x \right)} = x + 1.

    To find ddxf(x)\frac{d}{d x} f{\left(x \right)}:

    1. The derivative of log⁡(x)\log{\left(x \right)} is 1x\frac{1}{x}.

    To find ddxg(x)\frac{d}{d x} g{\left(x \right)}:

    1. Differentiate x+1x + 1 term by term:

      1. The derivative of the constant 11 is zero.

      2. Apply the power rule: xx goes to 11

      The result is: 11

    Now plug in to the quotient rule:

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

  2. Now simplify:

    −xlog⁡(x)+x+1x(x+1)2\frac{- x \log{\left(x \right)} + x + 1}{x \left(x + 1\right)^{2}}

The answer is:

−xlog⁡(x)+x+1x(x+1)2\frac{- x \log{\left(x \right)} + x + 1}{x \left(x + 1\right)^{2}}

The graph
02468-8-6-4-2-101020-10
Plot the graph f(x)
Plot the graph f'(x)
The first derivative [src] 
    1        log(x) 
--------- - --------
x*(x + 1)          2
            (x + 1)
−log⁡(x)(x+1)2+1x(x+1)- \frac{\log{\left(x \right)}}{\left(x + 1\right)^{2}} + \frac{1}{x \left(x + 1\right)}
Simplify
The second derivative [src] 
  1        2       2*log(x)
- -- - --------- + --------
   2   x*(1 + x)          2
  x                (1 + x) 
---------------------------
           1 + x
2log⁡(x)(x+1)2−2x(x+1)−1x2x+1\frac{\frac{2 \log{\left(x \right)}}{\left(x + 1\right)^{2}} - \frac{2}{x \left(x + 1\right)} - \frac{1}{x^{2}}}{x + 1}
Simplify
The third derivative [src] 
2    6*log(x)       3            6     
-- - -------- + ---------- + ----------
 3          3    2                    2
x    (1 + x)    x *(1 + x)   x*(1 + x) 
---------------------------------------
                 1 + x
−6log⁡(x)(x+1)3+6x(x+1)2+3x2(x+1)+2x3x+1\frac{- \frac{6 \log{\left(x \right)}}{\left(x + 1\right)^{3}} + \frac{6}{x \left(x + 1\right)^{2}} + \frac{3}{x^{2} \left(x + 1\right)} + \frac{2}{x^{3}}}{x + 1}
Simplify
The graph
Derivative of ln(x)/(x+1)
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