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.