Higher-Order Differentials

Differential of the second order of function $$$y={f{{\left({x}\right)}}}$$$ is differential of first differential of the function: $$${{d}}^{{2}}{y}={d}{\left({d}{y}\right)}$$$.

Differential of the third order of function $$$y={f{{\left({x}\right)}}}$$$ is differential of second differential of the function: $$${{d}}^{{3}}{y}={d}{\left({{d}}^{{2}}{y}\right)}$$$.

In general, differential of n-th order of function $$${y}={f{{\left({x}\right)}}}$$$ is differential of its $$${\left({n}-{1}\right)}-{t}{h}$$$ differential: $$${{d}}^{{n}}{y}={d}{\left({{d}}^{{{n}-{1}}}{y}\right)}$$$.

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

So, $$${{d}}^{{2}}{y}={d}{\left({d}{y}\right)}={d}{\left({y}'{d}{x}\right)}={d}{y}'{d}{x}={\left({y}''{d}{x}\right)}\cdot{d}{x}={y}''{d}{{x}}^{{2}}$$$,

$$${{d}}^{{3}}{y}={d}{\left({{d}}^{{2}}{y}\right)}={d}{\left({y}''{d}{{x}}^{{2}}\right)}={d}{y}''{d}{{x}}^{{2}}={\left({y}'''{d}{x}\right)}\cdot{d}{{x}}^{{2}}={y}'''{d}{{x}}^{{3}}$$$.

In general $$${\color{blue}{{{{d}}^{{{n}}}{y}={{y}}^{{{\left({n}\right)}}}{d}{{x}}^{{n}}}}}$$$.

From this we have that $$${{y}}^{{{\left({n}\right)}}}=\frac{{{{d}}^{{n}}{y}}}{{{d}{{x}}^{{n}}}}$$$.

From now we can treat this symbol as ratio.