Second derivative

The second derivative of a quadratic function is constant.

In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing.

On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with positive second derivative curves upwards, while the graph of a function with negative second derivative curves downwards.

Second derivative power rule

The power rule for the first derivative, if solved down a bit, will produce the second derivative power rule. The rule is given below:

$\frac{d^2}{dx^2}[x^n]=n(n-1)x^{(n-2)}=(n^2-n)x^{(n-2)}.$

Notation

For more details on this topic, see Notation for differentiation.

The second derivative of a function $f(x)\!$ is usually denoted $f''(x)\!$. That is:

$f'' = (f')'\!$

When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written

$\frac{d^2y}{dx^2}.$

This notation is derived from the following formula:

$\frac{d^2y}{dx^2} \,=\, \frac{d}{dx}\left(\frac{dy}{dx}\right).$

Example

Given the function

$f(x) = x^3,\!$

the derivative of f is the function

$f'(x) = 3x^2.\!$

The second derivative of f is the derivative of f′, namely

$f''(x) = 6x.\!$

Relation to the graph

A plot of $f(x) = \sin(2x)$ from $-\pi/4$ to $5\pi/4$. The tangent line is blue where the curve is concave up, green where the curve is concave down, and red at the inflection points (0, $\pi$/2, and $\pi$).

Concavity

The second derivative of a function f measures the concavity of the graph of f. A function whose second derivative is positive will be concave up (sometimes referred to as convex), meaning that the tangent line will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (sometimes called simply “concave”), and its tangent lines will lie above the graph of the function.

Inflection points

Main article: Inflection point

If the second derivative of a function changes sign, the graph of the function will switch from concave down to concave up, or vice versa. A point where this occurs is called an inflection point. Assuming the second derivative is continuous, it must take a value of zero at any inflection point, although not every point where the second derivative is zero is necessarily a point of inflection.

Second derivative test

The relation between the second derivative and the graph can be used to test whether a stationary point for a function (i.e. a point where $f'(x)=0\!$) is a local maximum or a local minimum. Specifically,

• If $\ f^{\prime\prime}(x) < 0$ then $\ f$ has a local maximum at $\ x$.
• If $\ f^{\prime\prime}(x) > 0$ then $\ f$ has a local minimum at $\ x$.
• If $\ f^{\prime\prime}(x) = 0$, the second derivative test says nothing about the point $\ x$, a possible inflection point.

The reason the second derivative produces these results can be seen by way of a real-world analogy. Consider a vehicle that at first is moving forward at a great velocity, but with a negative acceleration. Clearly the position of the vehicle at the point where the velocity reaches zero will be the maximum distance from the starting position – after this time, the velocity will become negative and the vehicle will reverse. The same is true for the minimum, with a vehicle that at first has a very negative velocity but positive acceleration.

Limit

It is possible to write a single limit for the second derivative:

$f''(x) = \lim_{h \to 0} \frac{f(x+h) - 2f(x) + f(x-h)}{h^2}.$

The expression on the right can be written as a difference quotient of difference quotients:

$\frac{f(x+h) - 2f(x) + f(x-h)}{h^2} = \frac{\frac{f(x+h) - f(x)}{h} - \frac{f(x) - f(x-h)}{h}}{h}.$

This limit can be viewed as a continuous version of the second difference for sequences.

Please note that the existence of the above limit does not mean that the function $f$ has a second derivative. The limit above just gives a possibility for calculating the second derivative but does not provide a definition. As a counterexample look on the sign function $\sgn(x)$ which is defined through

$\sgn(x) = \begin{cases} -1 & \text{if } x < 0, \\ 0 & \text{if } x = 0, \\ 1 & \text{if } x > 0. \end{cases}$

The sign function is not continuous at zero and therefore the second derivative for $x=0$ does not exist. But the above limit exists for $x=0$:

\begin{align} \lim_{h \to 0} \frac{\sgn(0+h) - 2\sgn(0) + \sgn(0-h)}{h^2} &= \lim_{h \to 0} \frac{1 - 2\cdot 0 + (-1)}{h^2} \\ &= \lim_{h \to 0} \frac{0}{h^2} \\ &= 0 \end{align}

Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function f. This is the quadratic function whose first and second derivatives are the same as those of f at a given point. The formula for the best quadratic approximation to a function f around the point x = a is

$f(x) \approx f(a) + f'(a)(x-a) + \tfrac12 f''(a)(x-a)^2.$

This quadratic approximation is the second-order Taylor polynomial for the function centered at x = a.

Eigenvalues and eigenvectors of the second derivative

For many combinations of boundary conditions explicit formulas for eigenvalues and eigenvectors of the second derivative can be obtained. For example, assuming $x \in [0,L]$ and homogeneous Dirichlet boundary conditions, i.e., $v(0)=v(L)=0$, the eigenvalues are $\lambda_j = -\frac{j^2 \pi^2}{L^2}$ and the corresponding eigenvectors (also called eigenfunctions) are $v_j(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{j \pi x}{L}\right)$. Here, $v''_j(x) = \lambda_j v_j(x), \, j=1,\ldots,\infty.$

For other well-known cases, see the main article eigenvalues and eigenvectors of the second derivative.

Generalization to higher dimensions

The Hessian

Main article: Hessian matrix

The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function f:R3 → R, these include the three second-order partials

$\frac{\part^2 f}{\part x^2}, \; \frac{\part^2 f}{\part y^2}, \text{ and }\frac{\part^2 f}{\part z^2}$

and the mixed partials

$\frac{\part^2 f}{\part x \, \part y}, \; \frac{\part^2 f}{\part x \, \part z}, \text{ and }\frac{\part^2 f}{\part y \, \part z}.$

If the function's image and domain both have a potential, then these fit together into a symmetric matrix known as the Hessian. The eigenvalues of this matrix can be used to implement a multivariable analogue of the second derivative test. (See also the second partial derivative test.)

The Laplacian

Main article: Laplace operator

Another common generalization of the second derivative is the Laplacian. This is the differential operator $\nabla^2$ defined by

$\nabla^2 f = \frac{\part^2 f}{\part x^2}+\frac{\part^2 f}{\part y^2}+\frac{\part^2 f}{\part z^2}.$

The Laplacian of a function is equal to the divergence of the gradient.

References

Print

• Anton, Howard; Bivens, Irl; Davis, Stephen (February 2, 2005), Calculus: Early Transcendentals Single and Multivariable (8th ed.), New York: Wiley, ISBN 978-0-471-47244-5
• Apostol, Tom M. (June 1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra 1 (2nd ed.), Wiley, ISBN 978-0-471-00005-1
• Apostol, Tom M. (June 1969), Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications 1 (2nd ed.), Wiley, ISBN 978-0-471-00007-5
• Eves, Howard (January 2, 1990), An Introduction to the History of Mathematics (6th ed.), Brooks Cole, ISBN 978-0-03-029558-4
• Larson, Ron; Hostetler, Robert P.; Edwards, Bruce H. (February 28, 2006), Calculus: Early Transcendental Functions (4th ed.), Houghton Mifflin Company, ISBN 978-0-618-60624-5
• Spivak, Michael (September 1994), Calculus (3rd ed.), Publish or Perish, ISBN 978-0-914098-89-8
• Stewart, James (December 24, 2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7
• Thompson, Silvanus P. (September 8, 1998), Calculus Made Easy (Revised, Updated, Expanded ed.), New York: St. Martin's Press, ISBN 978-0-312-18548-0