Higher-Order Derivatives

Summary

Higher-order derivatives are derivatives of derivatives. The second derivative $f^{″}$ measures how the first derivative changes (concavity in one variable). Mixed partials of several variables are related by Clairaut’s theorem under continuity hypotheses.

Prerequisites

Derivatives, Partial Derivatives

Definition

For a single-variable function,

and inductively $f^{(n)}=\frac{d}{dx}{f}^{(n-1)}$ .

For $f(x,y)$ , second partials include $f_{xx}$ , $f_{yy}$ , and mixed partials $f_{xy}$ , $f_{yx}$ .

Conditions / Assumptions

• Each differentiation step requires differentiability of the previous derivative on the region of interest.
• Equality of mixed partials $f_{xy}=f_{yx}$ holds when those partials are continuous (Clairaut); see Clairaut's Theorem.

Worked Example

If $f(x)=x^3$ , then $f^{\prime }(x)=3x^2$ , $f^{″}(x)=6x$ , $f^{‴}(x)=6$ , and $f^{(n)}(x)=0$ for $n\ge 4$ .

If $g(x)=e^x$ , then $g^{(n)}(x)=e^x$ for every $n\ge 0$ .

For $f(x,y)=x^{2}y$ , $f_x=2xy$ , $f_{xy}=2x$ , $f_{yx}=2x$ .

Common Mistakes

• Claiming that the Newtonian potential $1/r$ is harmonic in two dimensions. In ${\mathbb{R}}^2\setminus \{0\}$ , $\ln r$ is harmonic (away from the origin), while $1/r$ is harmonic in three dimensions (Coulomb potential), not in 2D.
• Omitting mixed-partial continuity hypotheses when swapping differentiation order.

Connections

• Clairaut's Theorem, Maxima and Minima (Hessian uses second partials)
• Series: Taylor polynomials use $f^{(n)}$

References

Higher derivatives and notation appear in OpenStax Calculus Volume 1; mixed partials in Volume 3.^[1]