Higher-Order Differentials

Differential of the second order of function `y=f(x)` is differential of first differential of the function: `d^2y=d(dy)`.

Differential of the third order of function `y=f(x)` is differential of second differential of the function: `d^3y=d(d^2y)`.

In general, differential of n-th order of function `y=f(x)` is differential of its `(n-1)-th` differential: `d^ny=d(d^(n-1)y)`.

When we calculate differentials it is important to remember that `dx` is arbitrary and independent from `x` number. So, when we differentiate with respect to `x` we treat `dx` as constant.

So, `d^2y=d(dy)=d(y'dx)=dy'dx=(y''dx)*dx=y''dx^2`,

`d^3y=d(d^2y)=d(y''dx^2)=dy''dx=(y'''dx)*dx^2=y'''dx^3`.

In general `color(blue)(d^(n)y=y^((n))dx^n)`.

From this we have that `y^((n))=(d^n y)/(dx^n)`.

From now we can treat this symbol as ratio.