Higher-Order Derivative¶

Definition¶

The derivative $f'$ is itself a function. If $f'$ is differentiable, its derivative is the second derivative

and we may continue whenever the next derivative exists, obtaining $f'''\text{,}$ $f^{(4)}\text{,}$ and so on. These are the higher-order derivatives of $f\text{.}$

Let $I\subseteq \mathbb{R}$ be an open interval and let $k\geq 1$ be an integer. A function $f\colon I\to\mathbb{R}$ is $k$-times differentiable on $I$ if the derivatives $f'\text{,}$ $f''\text{,}$ $\ldots\text{,}$ $f^{(k)}$ exist at every point of $I\text{.}$ Recursively, $f^{(1)}=f'$ and $f^{(k)}=\left(f^{(k-1)}\right)'$ for $k\geq 2\text{;}$ we also set $f^{(0)}=f\text{.}$ A function with derivatives of every order is called infinitely differentiable or smooth.

Interpretation¶

Higher-order derivatives measure change of change: $f'$ is the rate of change of $f$, $f''$ is the rate of change of $f'$, and so on. If $f(t)$ is the position of an object at time $t$, then $f'(t)$ is its velocity and $f''(t)$ its acceleration.

Each curve is the slope of the one before it. Where the red curve $f'(x)=x^2-1$ crosses $0$, at $x=\pm1$ (dashed lines), the graph of $f(x)=\tfrac{1}{3}x^3-x$ is flat: these are its stationary points. The green line $f''(x)=2x$ is negative at $-1$, so $f'$ is decreasing there and $f$ has a local maximum; it is positive at $1$, where $f$ has a local minimum.

The second derivative tells how the graph bends: $f''>0$ on an interval makes $f$ convex there and $f''<0$ makes it concave. This is the basis of the second-order condition for extrema and of inflection points; in the figure the graph of $f$ changes from concave to convex at $0$, where $f''$ changes sign.

Two slips are common. The brackets in $f^{(n)}$ matter: $f^{(2)}=f''$ is the second derivative, while $f^2=f\cdot f$ is a power; for $f=\sin$$\sin$, $f^{(2)}=-\sin$ but $f^2=\sin^2$${\sin }^2$. And a twice differentiable function need not have a continuous second derivative; that stronger property is the differentiability class $C^2$. Higher derivatives return in Math III as the coefficients $f^{(k)}(a)/k!$ of the Taylor polynomial.

Key Results¶

• Higher-order derivatives are defined recursively.
• Smooth functions admit derivatives of all orders.
• Second derivatives are particularly important for shape analysis.

Examples¶

• If $f(x)=x^3\text{,}$ then $f''(x)=6x\text{.}$
• Polynomial functions have derivatives of all orders.

Prerequisites¶

• Derivative and Differentiability

Related Concepts¶

• Convexity and Concavity
• Taylor Series
• Hessian Matrix
• Inflection Point
• Second-Order Partial Derivative
• Twice Continuous Differentiability on $\mathbb{R}^n$

Sources¶

• Math I: Calculus · Lecture 9

Aliases: higher order derivative, n-th derivative Symbols: real numbers, subset

Last updated: 20 October 2026 03:02