# 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.