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 .
2 years, 9 months ago

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