Third derivative

In calculus, the third derivative of a function f(x) is defined as the derivative of its second derivative f''(x), denoted as f'''(x) or \frac{d^3 f}{dx^3}, and it measures the rate of change of the concavity of the function's graph.^[1] This higher-order derivative provides insight into how rapidly the curvature—captured by the second derivative—is itself changing at a given point, which can indicate the behavior near inflection points or the sharpness of transitions in the function's shape.^[1] Notation for the third derivative follows standard conventions in differential calculus, such as the prime notation f'''(x) for successive differentiation or the Leibniz notation \frac{d^3 y}{dx^3} for a function y = f(x).^[1] For example, if f(x) = 5x^3 - 3x^2 + 10x - 5, the first derivative is f'(x) = 15x^2 - 6x + 10, the second is f''(x) = 30x - 6, and the third is the constant f'''(x) = 30.^[1] In general, for polynomial functions of degree n, the third derivative (and higher) will simplify accordingly, becoming zero if the order exceeds the degree.^[1] In physics and engineering, particularly kinematics, the third derivative of position with respect to time—known as jerk (symbol j)—represents the rate of change of acceleration and has units of meters per second cubed (m/s³).^[2] Jerk is crucial for analyzing motion profiles in systems like vehicles and machinery, where limiting its magnitude (e.g., below 2 m/s³ for passenger comfort in trains) helps minimize discomfort and mechanical stress.^[2] Early research by General Motors, for instance, identified jerk as a key factor in ride quality during road testing, extending beyond mere acceleration control to optimize dynamic responses in automotive design.^[3] Applications also extend to aerospace, such as constraining jerk in the Hubble Space Telescope's mechanisms to protect sensitive instruments.^[2]

Mathematical Foundations

Definition and Computation

In single-variable calculus, the third derivative of a function f(x) is defined as the derivative of its second derivative, denoted f'''(x) or \frac{d^3 f}{dx^3}, which quantifies the rate of change of the concavity of the function as determined by the second derivative f''(x).^[1]^[4] To compute the third derivative analytically, differentiation is applied successively starting from the original function. First, obtain the first derivative f'(x) using standard rules such as the power rule or product rule; then differentiate f'(x) to yield the second derivative f''(x); finally, differentiate f''(x) to arrive at f'''(x). For instance, consider the cubic polynomial f(x) = x^3: the first derivative is f'(x) = 3x^2, the second derivative is f''(x) = 6x, and the third derivative is f'''(x) = 6.^[1] This process assumes f(x) is sufficiently differentiable, and for polynomials of degree n, the (n+1)-th derivative and higher are zero.^[1] In multivariable calculus, the third derivative extends to partial derivatives, including pure partials like \frac{\partial^3 f}{\partial x^3} (differentiating three times with respect to x) and mixed partials such as \frac{\partial^3 f}{\partial x \partial y \partial z} for a function f(x, y, z). These are computed by applying the partial differentiation operator sequentially to the lower-order partials, following the same rules as single-variable cases but holding other variables constant. Clairaut's theorem, extended to higher orders, states that if the relevant partial derivatives are continuous in a neighborhood of the point, then mixed partials of the same total order are equal regardless of differentiation sequence—for example, \frac{\partial^3 f}{\partial x \partial y \partial z} = \frac{\partial^3 f}{\partial z \partial x \partial y}.^[5] When analytical computation is infeasible, numerical approximations of the third derivative can be obtained using finite difference methods, which rely on function evaluations at discrete points. A common forward finite difference formula for the third derivative at x with step size h > 0 is given by

f'''(x) \approx \frac{f(x + 3h) - 3f(x + 2h) + 3f(x + h) - f(x)}{h^3},

with a truncation error of order O(h).^[6] This formula derives from the third-order Taylor expansion and is exact for cubic polynomials.^[6]

Notation and Symbols

The third derivative of a function f(x) is commonly denoted using several established notations in mathematics. In Leibniz notation, introduced in the late 17th century, it is expressed as \frac{d^3 y}{dx^3} for a function y = f(x), emphasizing the infinitesimal changes through repeated differentiation.^[7] This form highlights the variable of differentiation in the denominator, making it particularly useful for partial derivatives or when specifying the independent variable.^[8] In contrast, Lagrange notation, developed in the 18th century, uses primes to indicate successive derivatives, writing the third derivative as f'''(x).^[9] This compact functional notation, inspired by earlier ideas from Newton, is prevalent in pure mathematics for its simplicity in expressing higher-order derivatives without explicit variables.^[10] Newton's notation, originally fluxional and adapted for time derivatives, employs dots over the variable to denote differentiation with respect to time t. For position x(t), the first derivative (velocity) is \dot{x}, the second (acceleration) is \ddot{x}, and the third (jerk) is \dddot{x}.^[8] This overdot convention extends naturally to higher orders but is primarily used in physics and engineering contexts involving temporal rates of change.^[11] The historical evolution of these notations traces back to the independent development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the 1670s. Leibniz first proposed his fractional form, including dx and dy, in a 1675 manuscript, formalizing it in publications from 1684 onward to represent differentials and their ratios.^[12] By the late 18th century, Joseph-Louis Lagrange refined the notation in his 1797 treatise Théorie des fonctions analytiques, introducing the prime symbol f'(x) as a shorthand for the derivative, which was later extended to multiple primes for higher orders; this shift aimed to treat derivatives as operations on functions rather than ratios of infinitesimals.^[9] Newton's dot notation, detailed in his Principia Mathematica (1687) and later works, evolved from fluxions and gained traction in applied sciences for its alignment with physical motions.^[10] In physics, particularly kinematics, the third time derivative of position—known as jerk—is often denoted by \dddot{x} or the symbol j, with units of m/s³.^[11] This extends Newton's dot notation and is used to describe changes in acceleration, such as in vehicle dynamics or roller coaster design.^[13] In vector calculus, \nabla^3 f is not standard notation for the third derivative but may refer to a higher-order operator, such as components of the third partial derivatives tensor, depending on context; it is distinct from scalar third derivatives like f'''(x).^[14] Conventions for higher-order derivatives build on these notations, with the fourth derivative (snap or jounce) marked by quadruple dots \ddddot{x} in Newtonian form or f^{(4)}(x) in Lagrange notation, distinguishing it from the third.^[13] The following table compares notations for the first three derivatives of y = f(x) or position x(t):

Order	Description	Leibniz Notation	Lagrange Notation	Newton Notation (time t)
1	First derivative	\frac{dy}{dx} or \frac{dx}{dt}	f'(x)	\dot{x}
2	Second derivative	\frac{d^2 y}{dx^2} or \frac{d^2 x}{dt^2}	f''(x)	\ddot{x}
3	Third derivative	\frac{d^3 y}{dx^3} or \frac{d^3 x}{dt^3}	f'''(x)	\dddot{x}

Interpretations in Calculus

Role in Taylor Series Expansion

In the Taylor series expansion of a function f(x) around a point a, the third derivative plays a crucial role in providing a cubic approximation to the function's local behavior. Taylor's theorem states that if f is three times differentiable in an interval containing a and x, then

f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f'''(c)}{3!}(x - a)^3

for some c between a and x, where the third-order term \frac{f'''(c)}{3!}(x - a)^3 captures the leading nonlinear deviation beyond quadratic behavior, enabling more precise approximations for functions with inflection or rapid changes near a.^[15] To quantify the accuracy of truncating the expansion after the quadratic term, the Lagrange form of the remainder after the second derivative (or equivalently, the error bound using the third derivative for the next term) is given by R_2(x) = \frac{f'''(\xi)}{3!}(x - a)^3 for some \xi between a and x, but for approximations up to the third order, the remainder becomes R_3(x) = \frac{f^{(4)}(\xi)}{4!}(x - a)^4 for some \xi between a and x, assuming f is four times differentiable. This remainder term highlights how the third derivative indirectly bounds the error in lower-order approximations by influencing the cubic contribution, which is essential for estimating convergence in numerical methods or series summations.^[16] A concrete example is the Maclaurin series expansion of \sin(x) around a = 0, where f(x) = \sin(x), f'(x) = \cos(x), f''(x) = -\sin(x), and f'''(x) = -\cos(x), yielding the third-order term -\frac{x^3}{6} since f'''(0) = -1. This cubic term approximates the oscillatory nature of \sin(x) near the origin, improving accuracy for small x compared to linear or quadratic approximations. The Maclaurin series, as the special case of Taylor's theorem at a = 0, relies on the third derivative to reveal cubic or higher-order asymmetries in function behavior close to the origin, particularly for oscillatory functions like trigonometric ones or those with inflection points.^[15]

Higher-Order Derivative Tests

In calculus, higher-order derivative tests extend the first and second derivative tests to classify critical points and inflection points when lower-order derivatives are inconclusive, particularly relying on the third derivative when the second derivative vanishes. These tests leverage the Taylor series expansion around a point c where f'(c) = 0, approximating the function's behavior locally as f(x) \approx f(c) + \frac{f''(c)}{2!}(x-c)^2 + \frac{f'''(c)}{3!}(x-c)^3 + \cdots. If the second derivative f''(c) = 0, the third derivative f'''(c) determines the nature of the point by examining the sign and order of the leading non-zero term.^[1] The third derivative test for inflection points applies when f''(c) = 0. If f'''(c) \neq 0, then c is an inflection point, as the concavity of f changes at c. This follows from the Taylor expansion, where the cubic term \frac{f'''(c)}{6}(x-c)^3 dominates near c, causing the second derivative to change sign: for f'''(c) > 0, the function transitions from concave down to concave up, and vice versa if f'''(c) < 0. For example, consider f(x) = x^3: here, f''(x) = 6x, so f''(0) = 0, and f'''(x) = 6 > 0 at x=0, confirming an inflection point where concavity changes from down (for x < 0) to up (for x > 0).^[17]^[18] For refining the classification of extrema at critical points where f''(c) = 0, the higher-order derivative test examines successive derivatives until finding the first non-zero one beyond the first. Suppose the lowest order k \geq 2 with f^{(k)}(c) \neq 0; a brief proof sketch uses Taylor's theorem, showing that near c, f(x) - f(c) has the sign of \frac{f^{(k)}(c)}{k!}(x-c)^k. If k is even and f^{(k)}(c) > 0, then c is a local minimum; if f^{(k)}(c) < 0, a local maximum. If k is odd (such as k=3), c is neither a local extremum nor an inflection point in the extremum sense but indicates a horizontal inflection, as the function changes monotonicity without extremal behavior. Thus, when f''(c) = 0 and f'''(c) \neq 0, the odd order implies no local extremum.^[19]^[18] An illustrative example is f(x) = x^4: f'(x) = 4x^3, so f'(0) = 0; f''(x) = 12x^2, so f''(0) = 0; f'''(x) = 24x, so f'''(0) = 0. The third derivative vanishes, requiring the fourth: f^{(4)}(x) = 24 > 0, an even order, confirming a local minimum at x=0. This distinguishes cases where even lower orders suffice from those needing higher scrutiny, unlike odd-order scenarios that preclude extrema.^[1] These tests have limitations: if f'''(c) = 0, the analysis is inconclusive, necessitating higher derivatives or alternative methods like the first derivative test. For instance, functions like f(x) = x^5 at x=0 have f'''(0) = 0 and an odd fifth-order term, yielding neither extremum nor inflection in the concavity sense, highlighting the need for complete Taylor assessment.^[19]

Applications in Physics

Jerk in Kinematics

In kinematics, jerk is defined as the third time derivative of an object's position or the first time derivative of its acceleration, quantifying the rate at which acceleration changes. Mathematically, for one-dimensional motion, it is expressed as j(t) = \frac{d^3 x(t)}{dt^3} = \frac{da(t)}{dt}, where x(t) is position and a(t) is acceleration.^[13]^[20] This derivative builds sequentially on lower-order kinematic quantities: position x(t) yields velocity v(t) = \frac{dx(t)}{[dt](/page/DT)}, which in turn yields acceleration a(t) = \frac{d^2 x(t)}{[dt](/page/DT)^2}, and finally jerk j(t) = \frac{d^3 x(t)}{[dt](/page/DT)^3}. In three-dimensional space, jerk is a vector \vec{j}(t) = \frac{d^3 \vec{r}(t)}{[dt](/page/DT)^3}, where \vec{r}(t) is the position vector, allowing analysis of directional changes in motion.^[13]^[20] Physically, jerk measures the abruptness of changes in an object's motion, often perceived as a sudden "push" or "pull" due to varying forces, which can induce discomfort or stress in mechanical systems and human passengers. For instance, high jerk during vehicle acceleration contributes to sensations of unease by rapidly altering the forces felt by occupants. Its SI unit is meters per second cubed (m/s³), reflecting the dimensional progression from position (m) through velocity (m/s) and acceleration (m/s²).^[13]^[20]

Examples in Mechanical Systems

In elevator systems, jerk-limited trajectories are employed to minimize passenger discomfort during acceleration and deceleration phases. These profiles typically feature segments of constant jerk, where the jerk j(t) remains steady, allowing for smooth transitions in acceleration. For instance, during the ramp-up phase, assuming initial conditions of zero velocity and acceleration, the position x(t) can be derived by successive integration: acceleration a(t) = j t, velocity v(t) = \frac{1}{2} j t^2, and position x(t) = \frac{1}{6} j t^3. Such profiles ensure jerk values below 1.0 m/s³, which are perceived as comfortable by passengers and align with standards like ISO 8100-34:2021 limiting jerk to 1.2 m/s³, reducing motion sickness and structural vibrations in the elevator mechanism.^[21]^[22]^[23] In projectile motion under variable gravity, such as near Earth's surface where gravitational acceleration g varies with altitude according to g(r) = GM / r^2 (with GM as Earth's gravitational parameter and r as distance from the center), the third derivative—jerk—is non-zero due to the changing acceleration. For a falling object, jerk arises from the radial dependence of gravity, expressed as \mathbf{j} = \frac{d g}{dr} \cdot \frac{dr}{dt}, incorporating both position and velocity terms, unlike constant-g approximations where jerk vanishes. This effect becomes noticeable in high-altitude or precise orbital simulations but is typically small near the surface, on the order of fractions of m/s³ for typical velocities.^[24] Vehicle dynamics during acceleration, such as a 0-60 mph sprint, often involve a constant jerk phase to smoothly ramp up acceleration and avoid abrupt shifts that could unsettle passengers or strain components. Here, jerk j = \frac{da}{dt}, where a is acceleration. For example, to achieve 0-60 mph (approximately 26.8 m/s) in 4 seconds with an average acceleration of about 6.7 m/s², a constant jerk phase might ramp acceleration from 0 to a maximum of 10 m/s² over 0.5 seconds, yielding j = 20 m/s³ during that interval; the velocity in this phase would then be v(t) = \frac{1}{2} j t^2 + a_0 t + v_0, with subsequent constant acceleration to reach the target speed. This approach is common in electric vehicles for optimal torque delivery without excessive wear.^[25]^[26] Dimensionally, jerk has SI units of m/s³, representing length per time cubed, which quantifies how rapidly acceleration changes and influences energy dissipation in mechanical systems through impulsive forces during transitions. In sharp stops, such as emergency braking, jerk values can reach 10–15 m/s³, leading to higher peak forces and potential passenger discomfort or component stress; for instance, city bus emergency braking systems report jerks up to 16 m/s³, emphasizing the need for limited-jerk designs to manage energy transfer efficiently without excessive vibrations.^[27]

Applications in Geometry

Relations to Curvature

In differential geometry, the third derivative of a plane curve parameterized by arc length \mathbf{r}(s) plays a key role in describing the variation of curvature \kappa(s) = |\mathbf{r}''(s)|. The vector \mathbf{r}''(s) points in the direction of the principal normal \mathbf{N}(s), with magnitude \kappa(s), while \mathbf{r}'''(s) captures how this curvature evolves along the curve.^[28] Within the Frenet-Serret framework for plane curves, the unit tangent vector is \mathbf{T}(s) = \mathbf{r}'(s), satisfying \mathbf{T}'(s) = \kappa(s) \mathbf{N}(s) and \mathbf{N}'(s) = -\kappa(s) \mathbf{T}(s). Differentiating \mathbf{r}''(s) = \kappa(s) \mathbf{N}(s) with respect to s yields the decomposition

\mathbf{r}'''(s) = \frac{d\kappa}{ds} \mathbf{N}(s) - \kappa(s)^2 \mathbf{T}(s),

which separates the third derivative into its tangential component (proportional to \mathbf{T}(s)) and normal component (proportional to \mathbf{N}(s)). The rate of change of curvature is then given by the projection \frac{d\kappa}{ds} = \mathbf{r}'''(s) \cdot \mathbf{N}(s). For unsigned curvature, the absolute value \left| \mathbf{r}'''(s) \cdot \mathbf{N}(s) \right| may be used in certain contexts.^[29]^[28] This relation highlights the third derivative's role in quantifying curvature variation. For instance, in a circle of radius R parameterized as \mathbf{r}(s) = (R \cos(s/R), R \sin(s/R)), the curvature \kappa = 1/R is constant, so \frac{d\kappa}{ds} = 0 and \mathbf{r}'''(s) = -\kappa^2 \mathbf{T}(s), aligning the third derivative with the tangential direction.^[29] The osculating circle at \mathbf{r}(s_0), with radius $1/\kappa(s_0) and center \mathbf{r}(s_0) + (1/\kappa(s_0)) \mathbf{N}(s_0), approximates the curve up to second order by matching position, tangent, and curvature. The third derivative refines this approximation, as the Taylor expansion \mathbf{r}(s_0 + h) = \mathbf{r}(s_0) + h \mathbf{T}(s_0) + (h^2/2) \kappa(s_0) \mathbf{N}(s_0) + (h^3/6) \mathbf{r}'''(s_0) + O(h^4) includes a third-order term (h^3/6) (\frac{d\kappa}{ds}(s_0) \mathbf{N}(s_0) - \kappa(s_0)^2 \mathbf{T}(s_0)), revealing deviations due to changing curvature.^[28]

Torsion in Space Curves

In the geometry of space curves, torsion quantifies the extent to which a curve twists out of its local osculating plane, providing a measure of deviation from planarity. For a curve \mathbf{r}(s) parametrized by arc length s, the torsion \tau(s) is defined as \tau(s) = \frac{ (\mathbf{r}'(s) \times \mathbf{r}''(s)) \cdot \mathbf{r}'''(s) }{ \| \mathbf{r}'(s) \times \mathbf{r}''(s) \|^2 }, where \mathbf{r}'(s), \mathbf{r}''(s), and \mathbf{r}'''(s) are the first, second, and third derivatives with respect to s.^[30] This formula arises from the scalar triple product, capturing the third derivative's contribution to the curve's helical behavior, with the denominator normalizing by the square of the curvature term.^[30] The Frenet-Serret equations relate the derivatives of the Frenet frame vectors—tangent \mathbf{T}, principal normal \mathbf{N}, and binormal \mathbf{B}—to curvature \kappa and torsion \tau:

\begin{align*} \frac{d\mathbf{T}}{ds} &= \kappa \mathbf{N}, \\ \frac{d\mathbf{N}}{ds} &= -\kappa \mathbf{T} + \tau \mathbf{B}, \\ \frac{d\mathbf{B}}{ds} &= -\tau \mathbf{N}. \end{align*}

These equations are derived by decomposing the third derivative \mathbf{r}'''(s) in the Frenet frame: \mathbf{r}'''(s) = -\kappa^2 \mathbf{T} + \kappa' \mathbf{N} + \kappa \tau \mathbf{B}, where the \kappa \tau \mathbf{B} term isolates torsion's influence on the binormal direction.^[30] The torsion coefficient \tau thus appears explicitly in the evolution of \mathbf{B}, reflecting the rate at which the osculating plane rotates around the tangent.^[31] A classic example is the circular helix \mathbf{r}(t) = (\cos t, \sin t, t), a non-planar space curve with constant curvature and torsion. The first derivative is \mathbf{r}'(t) = (-\sin t, \cos t, 1), the second is \mathbf{r}''(t) = (-\cos t, -\sin t, 0), and the third is \mathbf{r}'''(t) = (\sin t, -\cos t, 0). Computing the cross product \mathbf{r}'(t) \times \mathbf{r}''(t) = (\sin t, -\cos t, 1) yields \| \mathbf{r}'(t) \times \mathbf{r}''(t) \| = \sqrt{2}, and the triple product (\mathbf{r}'(t) \times \mathbf{r}''(t)) \cdot \mathbf{r}'''(t) = 1. Thus, the torsion is the constant value \tau = \frac{1}{(\sqrt{2})^2} = \frac{1}{2}, confirming the helix's uniform twisting.^[32] Geometrically, torsion \tau = 0 indicates a planar curve, where the binormal remains fixed and the curve lies entirely in one plane; positive \tau corresponds to right-handed helical twisting, as in the standard helix, while negative \tau denotes left-handed twisting.^[31] This interpretation underscores torsion's role in distinguishing three-dimensional curve configurations beyond mere bending.^[30]

Applications in Other Fields

Economic Marginal Analysis

For a cubic total cost function of the form C(q) = \alpha + \beta q + \gamma q^2 + \delta q^3, the third derivative \frac{d^3 C}{dq^3} = 6\delta is constant. If \delta > 0, this indicates that the second derivative (slope of marginal cost) increases linearly, representing an accelerating change in marginal costs. In production functions with cubic terms, such as f(q) = \alpha_1 q + \alpha_2 q^2 + \alpha_3 q^3, the marginal product is f'(q) = \alpha_1 + 2\alpha_2 q + 3\alpha_3 q^2, the second derivative f''(q) = 2\alpha_2 + 6\alpha_3 q captures changes in marginal returns, and the third derivative f'''(q) = 6\alpha_3. A positive f'''(q) (i.e., \alpha_3 > 0) implies an increasing rate of change in marginal returns, potentially reflecting accelerating efficiency in early production stages before diminishing effects.^[33]

Engineering and Control Systems

In engineering and control systems, the third derivative of position with respect to time, known as jerk, plays a crucial role in ensuring smooth actuation and minimizing vibrations in dynamic systems. In feedback control loops, such as those employing proportional-integral-derivative (PID) controllers, jerk minimization is integrated to prevent overshoot and mechanical stress during trajectory tracking. For instance, optimal tuning frameworks adapt performance indices like integral square error (ISE) and integral time absolute error (ITAE) to align controller gains with minimum-jerk profiles, enabling simultaneous optimization across multiple joints in robotic systems via particle swarm optimization (PSO). This approach enhances tracking accuracy while reducing peak control inputs and energy consumption compared to traditional tuning methods.^[34] In vibration engineering, the third derivative of beam deflection is fundamental to analyzing structural dynamics under load, particularly in the Euler-Bernoulli beam theory. This theory models slender beams by relating the shear force V to the third spatial derivative of transverse deflection w(x), expressed as V(x) = EI \frac{d^3 w}{dx^3}, where E is the modulus of elasticity and I is the moment of inertia. This relation derives from the equilibrium of forces and moments, with the third derivative capturing shear distribution essential for predicting deflections and stresses in applications like bridge design and machinery components. In dynamic contexts, it informs the governing partial differential equation \rho A \frac{\partial^2 v}{\partial t^2} + EI \frac{\partial^4 v}{\partial x^4} = f(x,t), where boundary conditions often set \frac{\partial^3 v}{\partial x^3} = 0 for free ends to reflect zero shear. A practical example of jerk application arises in robotic arm trajectory planning, where limits on the third derivative ensure feasible motion profiles that avoid excessive wear on actuators. Engineers often employ trapezoidal acceleration profiles, characterized by phases of constant jerk j(t) = J (where J is a bounded constant), transitioning smoothly between acceleration plateaus to compute position, velocity, and acceleration via successive integrations. This method optimizes time and smoothness for redundant manipulators, such as 7-DOF arms, by minimizing the integral of squared jerk in a multi-stage optimization process, reducing vibrations by up to 66% in experimental validations.^[35] In modern applications, particularly 2020s autonomous vehicles, third derivative constraints are imposed to enhance passenger comfort by limiting jerk during maneuvers like lane changes and braking. Discomfort studies indicate jerk limits around 0.42 m/s³ for lateral motion at 50% discomfort probability and up to 0.9 m/s³ for seated occupants in short pulses.^[36] These constraints, informed by ergonomic research, are integrated into motion planning algorithms, prioritizing triangular jerk profiles over sinusoidal for reduced perceived intensity. While standards like ISO 2631-1 evaluate whole-body vibration through acceleration metrics, jerk thresholds are derived from separate motion comfort studies.^[36]