Waste less time on Facebook — follow Brilliant.
×

Derivative of peculiar function II

Welcome to the part II of derivative of peculiar functions , well .. we will be discussing only one function. It is a good one. Inspired by Michael Huang and Pi Han Goh . Let's dive right into it!

\[ \large f^{(0)}(x)_{(n)} = x^{x^{x^{x \hspace{2mm} \cdots \small\text{ upto n ' } x \text{ '} }}} \]

\[\large f^{(\text{Represents no. of times the func. is diff. w.r.t } x)}(x)_{(\text{Represents number of } x \text{ in the tower })} \hspace{1mm} \] \[ \hspace{1mm} n \in N \]

Now let's take natural log on both sides .

\[ \ln{(f^{(0)}(x)_{(n)})} = f^{(0)}(x)_{(n-1)} \ln{x} \]

\[ \dfrac{1}{f^{(0)}(x)_{(n)}} \cdot f^{(1)}(x)_{(n)} = f^{(1)}(x)_{(n-1)} \cdot \ln{x} + \dfrac{1}{x} \cdot f^{(0)}(x)_{(n-1)} \]

\[ \begin{align} f^{(1)}(x)_{(n)} & = f^{(0)}(x)_{(n)} \left(f^{(1)}(x)_{(n-1)} \ln{x} + \dfrac{1}{x}\cdot f^{(0)}(x)_{(n-1)}\right) \\ f^{(1)}(x)_{(n-1)} & = f^{(0)}(x)_{(n-1)} \left( f^{(1)}(x)_{(n-2)} \ln{x} + \dfrac{1}{x}\cdot f^{(0)}(x)_{(n-2)}\right) \hspace{5mm} \times f^{(0)}(x)_{(n)} (\ln{x}) \\ f^{(1)}(x)_{(n-2)} & = f^{(0)}(x)_{(n-2)} \left( f^{(1)}(x)_{(n-3)} \ln{x} + \dfrac{1}{x}\cdot f^{(0)}(x)_{(n-3)}\right) \hspace{5mm} \times f^{(0)}(x)_{(n)}\cdot f^{(0)}(x)_{(n-1)} (\ln{x})^2 \\ \vdots \\ f^{(1)}(x)_{(3)} & = f^{(0)}(x)_{(3)} \left( f^{(1)}(x)_{(2)} \ln{x} + \dfrac{1}{x}\cdot f^{(0)}(x)_{(2)}\right) \hspace{4mm} \times \left( f^{(0)}(x)_{(n)} f^{(0)}(x)_{(n-1)} \cdots f^{(0)}(x)_{(4)} \right) (\ln{x})^{n-3} \\ f^{(1)}(x)_{(2)} & = f^{(0)}(x)_{(2)} \left( f^{(1)}(x)_{(1)} \ln{x} + \dfrac{1}{x}\cdot f^{(0)}(x)_{(1)}\right) \hspace{4mm} \times \left( f^{(0)}(x)_{(n)} f^{(0)}(x)_{(n-1)} \cdots f^{(0)}(x)_{(4)} f^{(0)}(x)_{(3)} \right) (\ln{x})^{n-2} \\ f^{(1)}(x)_{(1)} & = f^{(0)}(x)_{(1)} \left( f^{(1)}(x)_{(0)} \ln{x} + \dfrac{1}{x}\cdot f^{(0)}(x)_{(0)}\right) \hspace{4mm} \times \left( f^{(0)}(x)_{(n)} f^{(0)}(x)_{(n-1)} \cdots f^{(0)}(x)_{(4)} f^{(0)}(x)_{(3)} f^{(0)}(x)_{(2)} \right) (\ln{x})^{n-1} \end{align} \]

Add all these equations :

\[ \begin{align} f^{(1)}(x)_{(n)} & = \dfrac{1}{x}f^{(0)}(x)_{(n)}\cdot f^{(0)}(x)_{(n-1)}+ \dfrac{1}{x}f^{(0)}(x)_{(n)}\cdot f^{(0)}(x)_{(n-1)}\cdot f^{(0)}(x)_{(n-2)} (\ln{x})+ \\ & +\dfrac{1}{x}f^{(0)}(x)_{(n)}\cdot f^{(0)}(x)_{(n-1)}\cdot f^{(0)}(x)_{(n-2)} \cdot f^{(0)}(x)_{(n-3)} (\ln{x})^2 \\ \vdots \\& +\dfrac{1}{x}\left( f^{(0)}(x)_{(n)} f^{(0)}(x)_{(n-1)} \cdots f^{(0)}(x)_{(4)} f^{(0)}(x)_{(3)} f^{(0)}(x)_{(2)} \right) (\ln{x})^{n-3} \\ & +\dfrac{1}{x}\left( f^{(0)}(x)_{(n)} f^{(0)}(x)_{(n-1)} \cdots f^{(0)}(x)_{(4)} f^{(0)}(x)_{(3)} f^{(0)}(x)_{(2)}f^{(0)}(x)_{(1)} \right) (\ln{x})^{n-2} \\ & +\dfrac{1}{x}\left( f^{(0)}(x)_{(n)} f^{(0)}(x)_{(n-1)} \cdots f^{(0)}(x)_{(4)} f^{(0)}(x)_{(3)} f^{(0)}(x)_{(2)} f^{(0)}(x)_{(1)}f^{(0)}(x)_{(0)} \right) (\ln{x})^{n-1} \end{align} \]

\[\large \boxed{ f^{(1)}(x)_{(n)} = \dfrac{1}{x}f^{(0)}(x)_{(n)}\cdot f^{(0)}(x)_{(n-1)}\left(1+ \displaystyle\sum_{r=1}^{n-1}\left( \prod_{i=n-r-1}^{n-2} f^{(0)}(x)_{(i)} \right) ( \ln{x} )^{r} \right) } \]

Let's Check for \( n = 2 \)

\[ f^{(1)}(x)_{(2)} = \dfrac{1}{x}f^{(0)}(x)_{(2)}\cdot f^{(0)}(x)_{(1)}\left(1+ \displaystyle\sum_{r=1}^{1}\left( \prod_{i=1-r}^{0} f^{(0)}(x)_{(i)} \right) ( \ln{x} )^{r} \right) \]

\[ \dfrac{d\left(x^x\right)}{dx} = \dfrac{1}{x} \cdot x^x \cdot x \left( 1 + \ln{x} \right) = x^x(1+\ln{x} ) \]

For \( n = 3 \)

\[ \dfrac{d\left(x^{x^x}\right)}{dx} = \dfrac{1}{x} \cdot x^{x^x} \cdot x^x \left( 1 + x\ln{x}+ x(\ln{x})^2 \right) \]

Master challenge : Can you do it for \( f^{(m)}(x)_{(n)} \), where \( m \) is a natural number?

Note by Sabhrant Sachan
8 months, 4 weeks ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

There are no comments in this discussion.

×

Problem Loading...

Note Loading...

Set Loading...