Category: Higher-Order Derivatives

Definition of Higher-Order Derivatives

If $$${f{{\left({x}\right)}}}$$$ is a differentiable function, its derivative $$${f{'}}{\left({x}\right)}$$$ is also a function, so it may have a derivative (either finite or not). This function is called the second derivative of $$${f{{\left({x}\right)}}}$$$, because it is the derivative of a derivative, and is denoted by $$${f{''}}$$$. So, $$${f{''}}={\left({f{'}}\right)}'$$$.

Formulas for Higher-Order Derivatives

In general, to find n-th derivative of function $$$y={f{{\left({x}\right)}}}$$$ we need to find all derivatives of previous orders. But sometimes it is possible to obtain expression for n-th derivative that depends on $$$n$$$ and doesn't contain previous derivatives.

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: $$${{d}}^{{3}}{y}={d}{\left({{d}}^{{2}}{y}\right)}$$$.

Parametric Differentiating

Sometimes we need to write derivatives with respect to $$$x$$$ through differentials of another variable $$$t$$$. In this case expression for derivatives will be more complex.

So, let's calculate differentials with respect to $$$t$$$, in other words $$$x$$$ is not an independent variable.