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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:

$$\lim_{h\rightarrow0}\log_{\frac{log^nx+h}{log^nx}}\frac{log^nf(x+h)}{log^nf(x)}=\frac{dy}{dx}\frac{x}{y}\prod_{k=1}^n\frac{logx}{logy}$$

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?

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:

$$\lim_{h\rightarrow0}\log_{\frac{log^nx+h}{log^nx}}\frac{log^nf(x+h)}{log^nf(x)}=\frac{dy}{dx}\frac{x}{y}\prod_{k=1}^n\frac{logx}{logy}$$

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?

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