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Similar expressions
logx+1/(logx)^2

The derivative of the function/ logx-1/(logx)^2

Derivative of logx-1/(logx)^2

The solution

You have entered [src]

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

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

d /              1   \
--|log(x) - 1*-------|
dx|              2   |
  \           log (x)/

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

Transform

Detail solution

Differentiate $\log{\left(x \right)} - 1 \cdot \frac{1}{\log{\left(x \right)}^{2}}$ term by term:
1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
2. The derivative of a constant times a function is the constant times the derivative of the function.
  1. Let $u = \log{\left(x \right)}^{2}$ .
  2. Apply the power rule: $\frac{1}{u}$ goes to $- \frac{1}{u^{2}}$
  3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}^{2}$ :
    1. Let $u = \log{\left(x \right)}$ .
    2. Apply the power rule: $u^{2}$ goes to $2 u$
    3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \log{\left(x \right)}$ :
      1. The derivative of $\log{\left(x \right)}$ is $\frac{1}{x}$ .
      The result of the chain rule is:
      
      $\frac{2 \log{\left(x \right)}}{x}$
    The result of the chain rule is:
    
    $- \frac{2}{x \log{\left(x \right)}^{3}}$
  So, the result is: $\frac{2}{x \log{\left(x \right)}^{3}}$
The result is: $\frac{1}{x} + \frac{2}{x \log{\left(x \right)}^{3}}$

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

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

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

Simplify

The second derivative [src]

 /       2         6   \ 
-|1 + ------- + -------| 
 |       3         4   | 
 \    log (x)   log (x)/ 
-------------------------
             2           
            x

$$- \frac{1 + \frac{2}{\log{\left(x \right)}^{3}} + \frac{6}{\log{\left(x \right)}^{4}}}{x^{2}}$$

Simplify

The third derivative [src]

  /       2         9         12  \
2*|1 + ------- + ------- + -------|
  |       3         4         5   |
  \    log (x)   log (x)   log (x)/
-----------------------------------
                  3                
                 x

$$\frac{2 \cdot \left(1 + \frac{2}{\log{\left(x \right)}^{3}} + \frac{9}{\log{\left(x \right)}^{4}} + \frac{12}{\log{\left(x \right)}^{5}}\right)}{x^{3}}$$

Simplify

The graph

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Derivative of logx-1/(logx)^2

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The solution