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

Derivative of (1+lnx)/x

The solution

You have entered [src]

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

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

(1 + log(x))/x

Detail solution

Apply the quotient rule, which is:

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

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

To find $\frac{d}{d x} f{\left(x \right)}$ :
1. Differentiate $\log{\left(x \right)} + 1$ term by term:
  1. The derivative of the constant $1$ is zero.
  2. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
  The result is: $\frac{1}{x}$
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:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

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

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

Simplify

The third derivative [src]

5 - 6*log(x)
------------
      4     
     x

$$\frac{5 - 6 \log{\left(x \right)}}{x^{4}}$$

Simplify

The graph

Derivative of (1+lnx)/x