# 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.