Mister Exam

Lang:

Derivative of (ln^2(x))/(x^3)

You have entered [src]

   2   
log (x)
-------
    3  
   x

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

log(x)^2/x^3

Detail solution

Apply the quotient rule, which is:

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

$f{\left(x \right)} = \log{\left(x \right)}^{2}$ and $g{\left(x \right)} = x^{3}$ .

To find $\frac{d}{d x} f{\left(x \right)}$ :
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:
  
  $\frac{2 \log{\left(x \right)}}{x}$
To find $\frac{d}{d x} g{\left(x \right)}$ :
1. Apply the power rule: $x^{3}$ goes to $3 x^{2}$
Now plug in to the quotient rule:

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

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2              
  3*log (x)   2*log(x)
- --------- + --------
       4           3  
      x         x*x

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

Simplify

The second derivative [src]

  /                    2   \
2*\1 - 7*log(x) + 6*log (x)/
----------------------------
              5             
             x

$$\frac{2 \left(6 \log{\left(x \right)}^{2} - 7 \log{\left(x \right)} + 1\right)}{x^{5}}$$

Simplify

The third derivative [src]

  /            2               \
2*\-12 - 30*log (x) + 47*log(x)/
--------------------------------
                6               
               x

$$\frac{2 \left(- 30 \log{\left(x \right)}^{2} + 47 \log{\left(x \right)} - 12\right)}{x^{6}}$$

Simplify