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!

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

f(Represents no. of times the func. is diff. w.r.t x)(x)(Represents number of x in the tower )\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} nN \hspace{1mm} n \in N

Now let's take natural log on both sides .

ln(f(0)(x)(n))=f(0)(x)(n1)lnx \ln{(f^{(0)}(x)_{(n)})} = f^{(0)}(x)_{(n-1)} \ln{x}

1f(0)(x)(n)f(1)(x)(n)=f(1)(x)(n1)lnx+1xf(0)(x)(n1) \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)}

f(1)(x)(n)=f(0)(x)(n)(f(1)(x)(n1)lnx+1xf(0)(x)(n1))f(1)(x)(n1)=f(0)(x)(n1)(f(1)(x)(n2)lnx+1xf(0)(x)(n2))×f(0)(x)(n)(lnx)f(1)(x)(n2)=f(0)(x)(n2)(f(1)(x)(n3)lnx+1xf(0)(x)(n3))×f(0)(x)(n)f(0)(x)(n1)(lnx)2f(1)(x)(3)=f(0)(x)(3)(f(1)(x)(2)lnx+1xf(0)(x)(2))×(f(0)(x)(n)f(0)(x)(n1)f(0)(x)(4))(lnx)n3f(1)(x)(2)=f(0)(x)(2)(f(1)(x)(1)lnx+1xf(0)(x)(1))×(f(0)(x)(n)f(0)(x)(n1)f(0)(x)(4)f(0)(x)(3))(lnx)n2f(1)(x)(1)=f(0)(x)(1)(f(1)(x)(0)lnx+1xf(0)(x)(0))×(f(0)(x)(n)f(0)(x)(n1)f(0)(x)(4)f(0)(x)(3)f(0)(x)(2))(lnx)n1 \begin{aligned} 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{aligned}

Add all these equations :

f(1)(x)(n)=1xf(0)(x)(n)f(0)(x)(n1)+1xf(0)(x)(n)f(0)(x)(n1)f(0)(x)(n2)(lnx)++1xf(0)(x)(n)f(0)(x)(n1)f(0)(x)(n2)f(0)(x)(n3)(lnx)2+1x(f(0)(x)(n)f(0)(x)(n1)f(0)(x)(4)f(0)(x)(3)f(0)(x)(2))(lnx)n3+1x(f(0)(x)(n)f(0)(x)(n1)f(0)(x)(4)f(0)(x)(3)f(0)(x)(2)f(0)(x)(1))(lnx)n2+1x(f(0)(x)(n)f(0)(x)(n1)f(0)(x)(4)f(0)(x)(3)f(0)(x)(2)f(0)(x)(1)f(0)(x)(0))(lnx)n1 \begin{aligned} 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{aligned}

f(1)(x)(n)=1xf(0)(x)(n)f(0)(x)(n1)(1+r=1n1(i=nr1n2f(0)(x)(i))(lnx)r)\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 n = 2

f(1)(x)(2)=1xf(0)(x)(2)f(0)(x)(1)(1+r=11(i=1r0f(0)(x)(i))(lnx)r) 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)

d(xx)dx=1xxxx(1+lnx)=xx(1+lnx) \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 n = 3

d(xxx)dx=1xxxxxx(1+xlnx+x(lnx)2) \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) f^{(m)}(x)_{(n)} , where m m is a natural number?

Note by Sabhrant Sachan
4 years, 5 months ago

No vote yet
1 vote

  Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

  • Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
  • Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
  • Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

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]( 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


There are no comments in this discussion.


Problem Loading...

Note Loading...

Set Loading...