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Second derivative

In , the second derivative of a f(x), denoted as f''(x) or \frac{d^2 y}{dx^2}, is defined as the of the first f'(x), representing the instantaneous rate of change of the of the line to the of f. This higher-order quantifies in physical contexts, such as the second of with respect to time giving velocity's rate of change, and applies the power rule for polynomials, where for f(x) = x^n, f''(x) = n(n-1)x^{n-2}. Geometrically, the sign of f''(x) determines the concavity of the : if f''(x) > 0, the is concave up (the line lies below the curve); if f''(x) < 0, it is concave down (the line lies above the curve); and if f''(x) = 0, it may signal a point of inflection where concavity changes, though not always, as in the case of f(x) = x^4 at x = 0. A key application of the second derivative is the second derivative test for classifying critical points where f'(x) = 0: if f''(x) > 0 at such a point, it indicates a local minimum; if f''(x) < 0, a local maximum; and if f''(x) = 0, the test is inconclusive and further analysis is needed. For example, consider f(x) = x^3 - 9x^2 + 15x - 7; the critical points are at x = 1 and x = 5, with f''(1) = -12 < 0 confirming a local maximum and f''(5) = 12 > 0 confirming a local minimum. Points of occur where f''(x) changes sign, often at of f''(x) = 0 if the second derivative is continuous, aiding in sketching accurate graphs by revealing changes. Beyond single-variable , the second extends to multivariable functions via partial derivatives, such as the , which generalizes concavity tests for optimization in higher dimensions, though its interpretation remains rooted in assessing the behavior of first derivatives.

Fundamentals

Definition

The second derivative of a f, denoted f''(x), is defined as the derivative of the first derivative f'(x). It quantifies the instantaneous rate of change of the slope of the function's , thereby indicating the concavity or at a given point. This builds on the concept of the first derivative, which represents the instantaneous rate of change of f(x) itself and is assumed to be understood here as a prerequisite. Formally, if the first derivative is given by the limit f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}, then the second derivative is the derivative of this expression: f''(x) = \lim_{h \to 0} \frac{f'(x+h) - f'(x)}{h}, provided the limit exists./03%3A_Derivatives/3.02%3A_The_Derivative_as_a_Function) This definition captures how the function's slope accelerates or decelerates, offering insight into the function's bending behavior beyond mere steepness. The concept of the second derivative emerged in the as part of the foundational development of . incorporated higher-order fluxions, including second-order ones, in his unpublished manuscript (written around 1671, published 1736), where dots denoted successive rates of change. Independently, utilized second-order differentials in his 1684 paper Nova methodus pro maximis et minimis, employing notation like dd to represent differences of differences. These contributions laid the groundwork, though the notation and formalization of higher derivatives were refined later by mathematicians such as in the late .

Notation

The second derivative of a function y = f(x) is commonly denoted using several established notations in calculus, each with historical origins and specific contexts of use. Leibniz notation, developed by Gottfried Wilhelm Leibniz in the late 17th century, expresses the second derivative as \frac{d^2 y}{dx^2}, where the superscript 2 indicates the order of differentiation with respect to the independent variable x. This form emphasizes the ratio of infinitesimal changes and extends naturally to higher-order derivatives, such as \frac{d^n y}{dx^n} for the nth derivative, making it particularly advantageous for multivariable calculus and when tracking the differentiation variable explicitly. Lagrange notation, introduced by in his 1797 work Théorie des fonctions analytiques, uses prime symbols to denote s, so the second is written as f''(x) or y'' for a twice-differentiable f. This compact prime notation, also known as f^{(2)}(x) for the second order in some extensions, is favored in and when working with s evaluated at specific points, as it avoids specifying the variable unless necessary and remains concise even for moderate-order s. Newton's notation, originated by Isaac Newton in his fluxion-based calculus around 1665–1671, employs dots above the variable to indicate time derivatives, with the second derivative denoted as \ddot{y} for a time-dependent quantity y(t). This dot notation is predominantly used in physics and engineering for denoting accelerations or higher temporal rates of change, such as in , where the independent variable is implicitly time, though it becomes less practical for orders beyond the second or third due to typesetting limitations. These notations are equivalent for the second derivative of y = f(x), often written interchangeably as y'' = f''(x) = \frac{d^2 f}{dx^2} = \frac{d^2 y}{dx^2}, with the choice depending on the context: Leibniz for relational clarity in applied settings, Lagrange primes for , and Newton's dots for time-based dynamics.

Computation

The power rule provides an efficient method for computing the second derivative of power functions, which are monomials of the form f(x) = x^n, where n is a . The first is given by f'(x) = n x^{n-1}, and applying the power rule again yields the second derivative f''(x) = n(n-1) x^{n-2}. This second derivative formula arises directly from successive applications of the first power rule. Starting with f(x) = x^n, the first differentiation reduces the exponent by 1 and multiplies by n, resulting in f'(x) = n x^{n-1}. Differentiating once more treats n as a coefficient and n-1 as the new exponent, producing f''(x) = n(n-1) x^{n-2}. For constant terms in a function, such as f(x) = c where c is a (equivalent to x^0), the first derivative is zero, and thus the second derivative is also zero. Similarly, for linear terms like f(x) = mx + b, the first derivative is the m, making the second derivative zero. The power rule extends to general polynomials by applying it term-by-term. For a cubic polynomial f(x) = a x^3 + b x^2 + c x + d, the first derivative is f'(x) = 3a x^2 + 2b x + c, and the second derivative simplifies to f''(x) = 6a x + 2b, as higher-order terms vanish. While effective for pure power functions and polynomials, the power rule alone applies only to these forms; composite functions require additional rules, such as the or , for accurate second derivatives.

Limit Definition

The second derivative of a f, denoted f''(x), is formally defined as the of the first f'(x). That is, f''(x) = \lim_{h \to 0} \frac{f'(x + h) - f'(x)}{h}, provided the limit exists. This expression arises directly from applying the limit definition of the derivative to f' at the point x. To derive this from the first derivative's limit definition, recall that f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}. Substituting into the second derivative formula yields a nested limit: f''(x) = \lim_{h \to 0} \frac{1}{h} \left( \lim_{k \to 0} \frac{f(x + h + k) - f(x + h)}{k} - \lim_{k \to 0} \frac{f(x + k) - f(x)}{k} \right). Under suitable conditions where the limits can be interchanged (such as when f is twice continuously differentiable), this simplifies to the standard form above. An equivalent symmetric form, often used for its computational symmetry and higher-order accuracy in approximations, is f''(x) = \lim_{h \to 0} \frac{f(x + h) + f(x - h) - 2f(x)}{h^2}. This symmetric difference quotient can be obtained by expanding f'(x + h) and f'(x - h) using the first definition and combining terms, or via expansion assuming sufficient smoothness. For the second derivative to exist at x, the function f must be twice differentiable there, meaning f' exists in a neighborhood of x and is itself differentiable at x. If f'' exists, then f' is continuous at x, but the converse does not hold; f' may be continuous without being differentiable. In computational contexts, where exact is impractical, the second derivative is often approximated using s. The central formula f''(x) \approx \frac{f(x + h) + f(x - h) - 2f(x)}{h^2} for small h > 0 provides a second-order accurate , with on the order of O(h^2), assuming f is sufficiently smooth. This method is widely used in for solving differential equations and simulating physical systems.

Examples

To illustrate the computation of second derivatives, the following examples demonstrate step-by-step differentiation for representative functions using standard rules such as the power rule and . For the simple f(x) = x^3, apply the power rule to find the first : f'(x) = 3x^2. Differentiating once more with the power rule gives the second : f''(x) = 6x. For the trigonometric function f(x) = \sin x, the first derivative is f'(x) = \cos x, using the standard derivative rule for sine. Successive yields the second derivative: f''(x) = -\sin x. The f(x) = e^x has first f'(x) = e^x, following the rule that the of e^x is itself. Differentiating again produces the second : f''(x) = e^x. For the composite function f(x) = (x^2 + 1)^2, first apply the chain rule: let u = x^2 + 1, so f(x) = u^2 and f'(x) = 2u \cdot u' = 2(x^2 + 1) \cdot 2x = 4x(x^2 + 1). For the second derivative, differentiate the product $4x(x^2 + 1) using the : f''(x) = 4[(x^2 + 1) + x \cdot 2x] = 4(x^2 + 1 + 2x^2) = 4(3x^2 + 1) = 12x^2 + 4.

Geometric Interpretation

Concavity

The concavity of a 's is determined by the sign of its second derivative. A f is up (also known as ) on an if f''(x) > 0 for all x in that , meaning the lies above its tangent lines. Conversely, f is down (also known as ) on an if f''(x) < 0 for all x in that , meaning the lies below its tangent lines./04%3A_Applications_of_Derivatives/4.05%3A_Concavity_and_Inflection_Points) Intuitively, a positive second derivative indicates that the slope of the tangent line (given by the first derivative f') is increasing as x advances, causing the graph to bend upward like a cup holding water. A negative second derivative means the slope is decreasing, resulting in a downward bend. This behavior reflects the second derivative's role as the rate of change of the first derivative, quantifying the acceleration of the function's values. To determine concavity over a domain, one constructs a sign chart for f''(x) by identifying critical points where f''(x) = 0 or f''(x) is undefined, then testing the sign of f''(x) in the resulting intervals. For example, if f''(x) = 6x, the sign changes at x = 0: f''(x) < 0 for x < 0 (concave down) and f''(x) > 0 for x > 0 (concave up). Intervals where f''(x) > 0 confirm concavity up for f(x), providing a systematic way to analyze the graph's without plotting./04%3A_Applications_of_Differentiation/4.04%3A_Concavity_and_Curvature)

Inflection Points

An inflection point of a f at x = c is a point where the second f''(c) = 0 and the concavity of the graph changes, meaning f''(x) changes sign at c. This sign change indicates a transition from concave up (where f''(x) > 0) to concave down (where f''(x) < 0), or vice versa. The condition f''(c) = 0 is necessary for an inflection point but not sufficient on its own; verification of the sign change in f''(x) across intervals around c is required to confirm the concavity switch. To identify potential inflection points, solve the equation f''(x) = 0 for roots, then test the sign of f''(x) in the intervals determined by those roots, such as by evaluating at test points or using the first derivative test on f''(x). For example, consider f(x) = x^3. The second derivative is f''(x) = 6x, which equals zero at x = 0. Testing intervals shows f''(x) < 0 for x < 0 (concave down) and f''(x) > 0 for x > 0 (concave up), confirming an at x = 0 where concavity changes from down to up. Inflection points are significant because they mark locations where the graph's bending direction reverses, providing insight into the overall shape and behavior of the function beyond local extrema.

Relation to the Graph

The second derivative of a f(x), denoted f''(x), describes the rate of change of the of the line to the , thereby indicating how the bends relative to that . A positive value of f''(x) implies that the line is rotating counterclockwise as x increases, causing the to curve upward away from the . In contrast, a negative f''(x) results in rotation of the , leading the to bend downward toward the . This bending behavior manifests visually in the shape of the : when f''(x) > 0, the adopts a up form, often likened to a "smile" or the bottom of a U-shape, where the lies above its lines. Conversely, f''(x) < 0 produces a down appearance, resembling a "frown" or inverted U, with the positioned below the tangents. These qualitative features assist in sketching by highlighting regions where the accelerates or decelerates in its directional change./04%3A_Applications_of_Derivatives/4.05%3A_Concavity_and_Inflection_Points) The second derivative integrates with the first derivative f'(x) to reveal how the instantaneous slope evolves along the graph; specifically, the sign of f''(x) determines whether the slope is increasing (positive f'') or decreasing (negative f''), which directly influences the overall curvature and trajectory of the curve. For a quantitative measure of this curvature in the plane for y = f(x), the formula is \kappa(x) = \frac{|f''(x)|}{[1 + (f'(x))^2]^{3/2}}, where the absolute value of the second derivative scales the bending intensity, moderated by the slope's magnitude to account for the curve's orientation. This expression underscores the second derivative's central role in geometric interpretation, particularly when slopes are gentle, as the denominator approaches 1 and \kappa(x) \approx |f''(x)|.

Analytical Applications

Second Derivative Test

The second derivative test provides a method to classify critical points of a twice-differentiable f as local maxima or minima by evaluating the sign of the second derivative at those points. To apply the test, first identify the critical points by solving f'(c) = 0, assuming f''(c) exists. Then, evaluate f''(c): if f''(c) > 0, the function has a local minimum at c; if f''(c) < 0, it has a local maximum at c; and if f''(c) = 0, the test is inconclusive, requiring alternative methods such as the first derivative test. Consider the function f(x) = x^3 - 3x. The first derivative is f'(x) = 3x^2 - 3 = 3(x^2 - 1), so the critical points are at x = \pm 1. The second derivative is f''(x) = 6x. At x = -1, f''(-1) = -6 < 0, indicating a local maximum. At x = 1, f''(1) = 6 > 0, indicating a local minimum. The test fails to classify points when f''(c) = 0, as this case may correspond to a local extremum, a point of inflection, or neither. For instance, with f(x) = x^3, the critical point is at x = 0 where f'(0) = 0 and f''(0) = 0, but the function has neither a local maximum nor minimum there—instead, it is a point of inflection. Unlike the first derivative test, which examines the sign change of f' around the critical point and always provides a conclusion for differentiable functions, the second derivative test is often faster but limited to cases where the second derivative is nonzero.

Quadratic Approximation

The second-order Taylor expansion, also known as the quadratic approximation, provides a local representation of a f around a point a by incorporating the function's value, first derivative, and second derivative at that point. This approximation is given by f(x) \approx f(a) + f'(a)(x - a) + \frac{1}{2} f''(a) (x - a)^2, where the term \frac{1}{2} f''(a) (x - a)^2 captures the of the near a, with the sign and magnitude of f''(a) determining whether the parabola opens upward or downward. To account for the approximation's accuracy, includes a remainder term in Lagrange form: R_2(x) = \frac{f'''(\xi)}{3!} (x - a)^3, where \xi is some point between a and x, quantifying the error from neglecting higher-order terms. This form arises from applying repeatedly to the error function, ensuring the bound depends on the third derivative's maximum value in the interval. The approximation excels over the linear (first-order) Taylor polynomial for functions with significant , such as near maxima or minima, where the second derivative provides essential refinement—for instance, approximating \sin x near x = 0 yields \sin x \approx x - \frac{1}{6} x^3, but the second-order form \sin x \approx x improves locally only if is minimal, highlighting the quadratic term's role in curved regions. Geometrically, this parabola is to the of f at x = a, matching not only the function value and but also the concavity, thus providing a second-order contact that visually represents the function's bend.

Advanced Topics

Eigenvalues and Eigenvectors

In , the second derivative operator D^2 acts on a suitable , such as C^2[a, b] or Sobolev spaces, defined by D^2 f = f''. The associated eigenvalue problem is D^2 \phi = \lambda \phi, where \phi are eigenfunctions and \lambda are eigenvalues, typically negative for bounded domains to ensure real solutions. This linear operator is under appropriate boundary conditions, leading to real eigenvalues and orthogonal eigenfunctions, as established in Sturm-Liouville theory. The of D^2 depends strongly on the boundary conditions imposed on the [a, b]. For Dirichlet boundary conditions \phi(a) = \phi(b) = 0, the equation \phi'' = \lambda \phi yields eigenvalues \lambda_k = -(k \pi / (b - a))^2 and eigenfunctions \phi_k(x) = \sin(k \pi (x - a)/(b - a)) for k = 1, 2, \dots, assuming the interval length is scaled to [0, \pi] for simplicity. For Neumann conditions \phi'(a) = \phi'(b) = 0, the eigenvalues include a zero mode with \lambda_0 = 0 and \phi_0(x) = 1, followed by \lambda_k = -(k \pi / (b - a))^2 with \phi_k(x) = \cos(k \pi (x - a)/(b - a)) for k = 1, 2, \dots. \phi(a) = \phi(b) and \phi'(a) = \phi'(b) produce eigenvalues \lambda_k = -(2 \pi k / (b - a))^2 for integer k, with eigenfunctions comprising sines and cosines: \sin(2 \pi k (x - a)/(b - a)) and \cos(2 \pi k (x - a)/(b - a)). These conditions alter the by excluding or including certain modes, with Dirichlet enforcing stricter decay and periodic allowing wave-like propagation. In numerical methods, the second derivative is discretized via finite differences on a with n interior points and spacing h = (b - a)/(n+1), yielding the approximating D^2 with 1 on the sub- and superdiagonals and -2 on the , scaled by $1/h^2. For Dirichlet boundaries, the eigenvalues of this matrix are \lambda_k = -\frac{4}{h^2} \sin^2 \left( \frac{k \pi}{2(n+1)} \right) for k = 1, \dots, n, with corresponding eigenvectors having components \sin(j k \pi / (n+1)) for grid index j. Neumann or periodic discretizations modify the matrix (e.g., adjusting corner entries for periodic), shifting the —Neumann includes a near-zero eigenvalue, while periodic uses circulant structure with eigenvalues \lambda_k = -\frac{4}{h^2} \sin^2 \left( \frac{k \pi}{n} \right). These discrete spectra approximate the continuous ones as n \to \infty, converging to the exact eigenvalues. Mathematically, the eigenstructure of D^2 underpins expansions like , where eigenfunctions form complete orthogonal bases for solving PDEs. In applications such as , the time-independent for a in a uses D^2 \psi = -\frac{2mE}{\hbar^2} \psi with Dirichlet conditions, yielding discrete energy levels E_k \propto k^2; similarly, in vibration theory, normal modes of a satisfy D^2 \phi = -\omega^2 \phi under periodic conditions, determining frequencies \omega_k \propto |k|. These frameworks emphasize the operator's role in rather than physical details.

Higher Dimensions

In , the second derivative generalizes to of several variables through . For a function f(x, y) of two variables, the second-order partial derivatives include the pure second partials f_{xx} = \frac{\partial^2 f}{\partial x^2} and f_{yy} = \frac{\partial^2 f}{\partial y^2}, as well as the mixed partial f_{xy} = \frac{\partial^2 f}{\partial x \partial y}. These mixed partials measure the rate of change of the first partial derivative with respect to the other variable, capturing interactions between variables. A key property is the equality of mixed partial derivatives, established by Clairaut's theorem: if f_{xy} and f_{yx} are both continuous on a disk containing the point (a, b), then f_{xy}(a, b) = f_{yx}(a, b). This symmetry holds under the continuity assumption, simplifying computations by allowing the order of differentiation to be interchanged. For functions of n variables, \mathbf{x} = (x_1, \dots, x_n), the second partial derivatives take the general form \frac{\partial^2 f}{\partial x_i \partial x_j} for i, j = 1, \dots, n, yielding n^2 such derivatives in total. The one-variable second derivative corresponds to the special case where n=1 and i = j = 1. By the generalization of Clairaut's theorem to multiple variables, if the relevant mixed partials are continuous in a neighborhood, then \frac{\partial^2 f}{\partial x_i \partial x_j} = \frac{\partial^2 f}{\partial x_j \partial x_i}, making the collection of second partials a . Higher-order partial derivatives extend this framework, such as third-order partials like \frac{\partial^3 f}{\partial x^3} or mixed forms like \frac{\partial^3 f}{\partial x^2 \partial y}, though the focus remains on second-order for analyzing local behavior. The equality of mixed higher-order partials follows similarly from continuity conditions, generalizing to arbitrary orders.

Hessian Matrix

In multivariable calculus, the Hessian matrix of a scalar-valued function f: \mathbb{R}^n \to \mathbb{R} is the n \times n symmetric of second-order partial derivatives, with entries given by H_{ij} = \frac{\partial^2 f}{\partial x_i \partial x_j}. The symmetry arises because mixed partial derivatives are equal under sufficient conditions, so H_{ij} = H_{ji}. For a function of two variables f(x, y), the Hessian matrix takes the form H = \begin{bmatrix} \frac{\partial^2 f}{\partial x^2} & \frac{\partial^2 f}{\partial x \partial y} \\ \frac{\partial^2 f}{\partial y \partial x} & \frac{\partial^2 f}{\partial y^2} \end{bmatrix}, often denoted as \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix}. This structure generalizes the second derivative from one dimension to higher dimensions, capturing the local of the . At a critical point where the gradient \nabla f = 0, the eigenvalues of the Hessian determine the nature of the extremum: if all eigenvalues are positive (positive definite Hessian), the point is a local minimum; if all are negative (negative definite), it is a local maximum; if they have mixed signs (indefinite), it is a saddle point. If the Hessian is positive semidefinite (nonnegative eigenvalues, at least one zero), the test is inconclusive for a minimum, and similarly for negative semidefinite cases. This eigenvalue-based classification extends the second derivative test from single-variable calculus, where a positive second derivative indicates a local minimum. In optimization, the plays a central role in methods like Newton's algorithm, where the update step from a current point x_t is x_{t+1} = x_t - H^{-1}(x_t) \nabla f(x_t), with H(x_t) being the at x_t. This uses the inverse to precondition the , accounting for the function's and enabling quadratic near local minima when the is positive definite.

Laplacian

The Laplacian operator, denoted by Δ, applied to a twice-differentiable scalar f in n-dimensional , is defined as the of the second partial derivatives with respect to each coordinate: \Delta f = \sum_{i=1}^n \frac{\partial^2 f}{\partial x_i^2}. This arises from the need to quantify isotropic second-order changes in the across all directions. Equivalently, the Laplacian is the trace of the of f, which captures the of the principal curvatures in a coordinate-independent manner. In specific dimensions, the expression simplifies accordingly. For a function of two variables, \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2}, while in three variables, it extends to \Delta f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}. These forms highlight the operator's role in measuring average without directional bias. The Laplacian possesses key invariance properties: it remains unchanged under rotations and translations of the , ensuring its applicability across different frames. Algebraically, it can be expressed as the of the : \Delta f = \nabla \cdot (\nabla f), which underscores its foundations and facilitates its use in theorems. A central application is , \Delta f = 0, whose solutions are harmonic functions. These functions exhibit the mean value property—where the value at any point equals the average over any surrounding ball—and are infinitely differentiable, reflecting maximal smoothness among solutions to elliptic partial differential equations. , \Delta f = g, extends this framework to include a source term g, modeling inhomogeneous phenomena such as gravitational or electromagnetic potentials driven by distributed sources. In mathematical physics, the Laplacian drives fundamental models with an emphasis on analytical properties. The heat equation, \frac{\partial u}{\partial t} = \kappa \Delta u, uses the Laplacian to describe diffusive , where \kappa > 0 is the diffusion coefficient; steady-state solutions (\partial u / \partial t = 0) reduce to . Similarly, the wave equation, \frac{\partial^2 u}{\partial t^2} = c^2 \Delta u, incorporates the Laplacian for propagation speeds c, with stationary yielding functions. In , the \phi satisfies \Delta \phi = 0 in regions without charge () and \Delta \phi = -\rho / \epsilon_0 with \rho (), enabling exact solutions via in symmetric domains.