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Derivative of:
Derivative of x^-(4/5)
Derivative of x^2*tgx
Derivative of ln(2x)/x
Derivative of acos(t)
Identical expressions
y=ln^2x/x^ five =ctg5x
y equally ln squared x divide by x to the power of 5 equally ctg5x
y equally ln squared x divide by x to the power of five equally ctg5x
y=ln2x/x5=ctg5x
y=ln²x/x⁵=ctg5x
y=ln to the power of 2x/x to the power of 5=ctg5x
y=ln^2x divide by x^5=ctg5x

The derivative of the function/ y=ln^2x/x^5=ctg5x

Derivative of y=ln^2x/x^5=ctg5x

The solution

You have entered [src]

   2   
log (x)
-------
    5  
   x

$$\frac{\log{\left(x \right)}^{2}}{x^{5}}$$

log(x)^2/x^5

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 $g{\left(x \right)} = x^{5}$ .

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: $x^{5}$ goes to $5 x^{4}$
Now plug in to the quotient rule:

$\frac{- 5 x^{4} \log{\left(x \right)}^{2} + 2 x^{4} \log{\left(x \right)}}{x^{10}}$
Now simplify:

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

The answer is:

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

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

       2              
  5*log (x)   2*log(x)
- --------- + --------
       6           5  
      x         x*x

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

Simplify

The second derivative [src]

  /                      2   \
2*\1 - 11*log(x) + 15*log (x)/
------------------------------
               7              
              x

$$\frac{2 \left(15 \log{\left(x \right)}^{2} - 11 \log{\left(x \right)} + 1\right)}{x^{7}}$$

Simplify

The third derivative [src]

  /             2                \
2*\-18 - 105*log (x) + 107*log(x)/
----------------------------------
                 8                
                x

$$\frac{2 \left(- 105 \log{\left(x \right)}^{2} + 107 \log{\left(x \right)} - 18\right)}{x^{8}}$$

Simplify

The graph

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Derivative of y=ln^2x/x^5=ctg5x

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The solution