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Identical expressions
е^(-lnx^ three)^ five
е to the power of ( minus lnx cubed ) to the power of 5
е to the power of ( minus lnx to the power of three) to the power of five
е(-lnx3)5
е-lnx35
е^(-lnx³)⁵
е to the power of (-lnx to the power of 3) to the power of 5
е^-lnx^3^5
Similar expressions
е^(lnx^3)^5

The derivative of the function/ е^(-lnx^3)^5

Derivative of е^(-lnx^3)^5

The solution

You have entered [src]

 /          5\
 |/    3   \ |
 \\-log (x)/ /
e

e^{\left(- \log{\left(x \right)}^{3}\right)^{5}}

  / /          5\\
  | |/    3   \ ||
d | \\-log (x)/ /|
--\e             /
dx

\frac{d}{d x} e^{\left(- \log{\left(x \right)}^{3}\right)^{5}}

Transform

Detail solution

Let $u = \left(- \log{\left(x \right)}^{3}\right)^{5}$ .
The derivative of $e^{u}$ is itself.
Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \log{\left(x \right)}^{3}\right)^{5}$ :
1. Let $u = - \log{\left(x \right)}^{3}$ .
2. Apply the power rule: $u^{5}$ goes to $5 u^{4}$
3. Then, apply the chain rule. Multiply by $\frac{d}{d x} \left(- \log{\left(x \right)}^{3}\right)$ :
  1. 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. Apply the power rule: $u^{3}$ goes to $3 u^{2}$
    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:
      
      $\frac{3 \log{\left(x \right)}^{2}}{x}$
    So, the result is: $- \frac{3 \log{\left(x \right)}^{2}}{x}$
  The result of the chain rule is:
  
  $- \frac{15 \log{\left(x \right)}^{14}}{x}$
The result of the chain rule is:

$- \frac{15 e^{\left(- \log{\left(x \right)}^{3}\right)^{5}} \log{\left(x \right)}^{14}}{x}$
Now simplify:

$- \frac{15 e^{- \log{\left(x \right)}^{15}} \log{\left(x \right)}^{14}}{x}$

The answer is:

- \frac{15 e^{- \log{\left(x \right)}^{15}} \log{\left(x \right)}^{14}}{x}

The graph

Plot the graph f(x)

Plot the graph f'(x)

The first derivative [src]

              /          5\
              |/    3   \ |
       14     \\-log (x)/ /
-15*log  (x)*e             
---------------------------
             x

- \frac{15 e^{\left(- \log{\left(x \right)}^{3}\right)^{5}} \log{\left(x \right)}^{14}}{x}

Simplify

The second derivative [src]

                                          /          5\
                                          |/    3   \ |
      13    /            15            \  \\-log (x)/ /
15*log  (x)*\-14 + 15*log  (x) + log(x)/*e             
-------------------------------------------------------
                            2                          
                           x

\frac{15 \cdot \left(15 \log{\left(x \right)}^{15} + \log{\left(x \right)} - 14\right) e^{\left(- \log{\left(x \right)}^{3}\right)^{5}} \log{\left(x \right)}^{13}}{x^{2}}

Simplify

The third derivative [src]

                                                                                        /          5\
                                                                                        |/    3   \ |
      12    /              30            16           2                         15   \  \\-log (x)/ /
15*log  (x)*\-182 - 225*log  (x) - 45*log  (x) - 2*log (x) + 42*log(x) + 630*log  (x)/*e             
-----------------------------------------------------------------------------------------------------
                                                   3                                                 
                                                  x

\frac{15 \left(- 225 \log{\left(x \right)}^{30} - 45 \log{\left(x \right)}^{16} + 630 \log{\left(x \right)}^{15} - 2 \log{\left(x \right)}^{2} + 42 \log{\left(x \right)} - 182\right) e^{\left(- \log{\left(x \right)}^{3}\right)^{5}} \log{\left(x \right)}^{12}}{x^{3}}

Simplify

The graph

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Derivative of е^(-lnx^3)^5

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The solution