1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

B Defining a new kind of derivative

  1. Feb 2, 2017 #1
    I'm using these notations:

    1.$log^n_xy$: For log with the base $x$ applied $n$ times to $y$. For example, $log^3y=log(log(log(y))$ all with the same base.

    2.$^{n[x]}a$: For the power tower or repeated exponentiation of $a$ evaluated from right to left such that the number of $a$'s in the operation is $n$ and it has $x$ on the top. For example, $$^{3[5]}2=2^{2^{2^5}}$$, $$^{0[8]}2=8$$ The power tower contains no $2$'s in the second case.

    I've defined this new kind of derivative:


    where $\frac{dy}{dx}$ is the normal derivative of $f(x)$ and all the logarithms whose base is unspecified should be calculated to the the same base $a$.

    I've not found any use of this derivative except in approximations, in which it is better than the normal derivative. I've got these approximation results:

    By taking the derivative at $n=0$,

    $$e^{x_2}\approx e^{x_1}(\frac{x_2}{x_1})^{x_1}$$,

    $$x_2^2+x_2\approx (x_1^2+x_1)(\frac{x_2}{x_1})^{\frac{2x_1+1}{x_1+1}}$$

    when $\frac{x_2}{x_1}\approx 1$

    $$^{(n)[kx]}e\approx ^{(n)[x]}e+\log_{e}k\prod_{i=0}^n{^i[x]}e$$, when $k\approx 1$

    $$\frac{\log_ax_2-\log_ax_1}{\log^{n+1}_ax_2-\log^{n+1}_ax_1}*\frac{1}{((ln(a))^n}\approx \prod_{i=1}^nlog_a^ix_1$$, when $\frac{\log_{a}^{n}x_2}{\log_{a}^{n}x_1}\approx 1$

    Can these results be proved by using existing maths concepts without using this derivative of mine?
    Last edited: Feb 2, 2017
  2. jcsd
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted