Xrfgtjtk

I have the following formula:

$$f(x) = frac{d^2w(x)}{dx^2}$$

Now I would like to normalize $x$ by dividing it by $L$? This would be the substitution: $$hat{x}=x/L$$

How would my formula change? (step by step please)

Using $x = L hat{x}$ we have $dx = L dhat{x}$ and $dx^2 = L^2 dhat{x}^2$. Therefore,
$$
frac{d}{dx^2} = frac{1}{L^2} frac{d}{dhat{x}^2}.
$$
This simple "substitution" is not mathematically rigorous, but you could use the chain rule twice to obtain the same thing; since the change of variable consists only in the multiplication by a constant, this "heuristic" works.

The final expression is
$$
f(L hat{x}) = frac{1}{L^2} frac{d}{dhat{x}^2} w(Lhat{x}).
$$

If, for example, $w(x) = A exp(Bx)$, we have $f(x) = AB^2 exp(Bx)$. With the new variable,
$$
A (BL)^2 exp(BL hat{x}) = frac{d}{dhat{x}^2} A exp(BL hat{x}).
$$
If you define $hat{B}=BL$, $hat{f} = hat{B}^2 exp(hat{B} hat{x})$ and $hat{w}=exp(hat{B} hat{x})$, the equation is
$$
hat{f} = frac{d hat{w}}{dhat{x}^2} .
$$
Therefore, after you normalize $x$ with $hat{x}$, you should also normalize the functions. If $w$ has units of Kelvin, for example, $f$ has units of $K/m^2$. After the normalization, both $hat{w}$ and $hat{f}$ are nondimensional.

Appendix: rigorous change of variable using chain rule

Let $x=L hat{x}$. Therefore, $hat{x}=x/L$. From the chain rule,
$$
frac{dw}{dx} = frac{dhat{x}}{dx} frac{dw}{dhat{x}} = frac{1}{L} frac{dw}{dhat{x}}
$$
and
$$
frac{d^2w}{dx^2}=frac{d}{dx} left(frac{dw}{dx}right) = frac{dhat{x}}{dx} frac{d}{dhat{x}}left(frac{1}{L} frac{dw}{dhat{x}} right) = frac{1}{L^2} frac{d^2 w}{d hat{x}^2}.
$$