Second derivative

Notation

The second derivative of a function ${\displaystyle f(x)}$ is usually denoted ${\displaystyle f^{″}(x)}$. That is: $${\displaystyle f^{″}={\left(f^{\prime }\right)}^{\prime }}$$ When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written $${\displaystyle \frac{d^{2}y}{d{x}^2}.}$$ This notation is derived from the following formula: $${\displaystyle \frac{d^{2}y}{d{x}^2}=\frac{d}{dx}\left(\frac{dy}{dx}\right).}$$

Example

Given the function $${\displaystyle f(x)=x^3,}$$ the derivative of f is the function $${\displaystyle f^{\prime }(x)=3x^2.}$$ The second derivative of f is the derivative of ${\displaystyle f^{\prime }}$, namely $${\displaystyle f^{″}(x)=6x.}$$

Relation to the graph

Concavity

The second derivative of a function f can be used to determine the concavity of the graph of f. A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function. Similarly, a function whose second derivative is negative will be concave down (sometimes simply called concave), and its tangent line will lie above the graph of the function near the point of contact.

Inflection points

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 ${\displaystyle f^{\prime }(x)=0}$) is a local maximum or a local minimum. Specifically,

• If ${\displaystyle f^{″}(x)<0}$, then ${\displaystyle f}$ has a local maximum at ${\displaystyle x}$.
• If ${\displaystyle f^{″}(x)>0}$, then ${\displaystyle f}$ has a local minimum at ${\displaystyle x}$.
• If ${\displaystyle f^{″}(x)=0}$, the second derivative test says nothing about the point ${\displaystyle x}$, 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: $${\displaystyle f^{″}(x)=\underset{h\to 0}{lim}\frac{f(x+h)-2f(x)+f(x-h)}{h^2}.}$$

The limit is called the second symmetric derivative. The second symmetric derivative may exist even when the (usual) second derivative does not.

The expression on the right can be written as a difference quotient of difference quotients: $${\displaystyle \frac{f(x+h)-2f(x)+f(x-h)}{h^2}=\frac{{\displaystyle \frac{f(x+h)-f(x)}{h}}-{\displaystyle \frac{f(x)-f(x-h)}{h}}}{h}.}$$ This limit can be viewed as a continuous version of the second difference for sequences.

However, the existence of the above limit does not mean that the function ${\displaystyle f}$ has a second derivative. The limit above just gives a possibility for calculating the second derivative—but does not provide a definition. A counterexample is the sign function ${\displaystyle \mathrm{sgn}(x)}$, which is defined as: $${\displaystyle \mathrm{sgn}(x)=\{\begin{array}{ll}-1& \text{if }x<0,\\ 0& \text{if }x=0,\\ 1& \text{if }x>0.\end{array}}$$

The sign function is not continuous at zero, and therefore the second derivative for ${\displaystyle x=0}$ does not exist. But the above limit exists for ${\displaystyle x=0}$: $${\displaystyle \begin{array}{cc}\underset{h\to 0}{lim}\frac{\mathrm{sgn}(0+h)-2\mathrm{sgn}(0)+\mathrm{sgn}(0-h)}{h^2}& =\underset{h\to 0}{lim}\frac{\mathrm{sgn}(h)-2\cdot 0+\mathrm{sgn}(-h)}{h^2}\\ & =\underset{h\to 0}{lim}\frac{\mathrm{sgn}(h)+(-\mathrm{sgn}(h))}{h^2}=\underset{h\to 0}{lim}\frac{0}{h^2}=0.\end{array}}$$

Quadratic approximation

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 $${\displaystyle f(x)\approx f(a)+f^{\prime }(a)(x-a)+\frac{1}{2}{f}^{″}(a)(x-a{)}^2.}$$ 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 ${\displaystyle x\in [0,L]}$ and homogeneous Dirichlet boundary conditions (i.e., ${\displaystyle v(0)=v(L)=0}$ where v is the eigenvector), the eigenvalues are ${\displaystyle {\lambda }_j=-\frac{j^2{\pi }^2}{L^2}}$ and the corresponding eigenvectors (also called eigenfunctions) are ${\displaystyle v_j(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{j\pi x}{L}\right)}$. Here, ${\displaystyle v_j^{″}(x)={\lambda }_j{v}_j(x)}$, for ${\displaystyle j=1,\dots ,\mathrm{\infty }}$.

For other well-known cases, see Eigenvalues and eigenvectors of the second derivative.

Generalization to higher dimensions

The Hessian

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 $${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2},\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2},\text{ and }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}}$$ and the mixed partials $${\displaystyle \frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }y},\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }x\mathrm{\partial }z},\text{ and }\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }y\mathrm{\partial }z}.}$$

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

Another common generalization of the second derivative is the Laplacian. This is the differential operator ${\displaystyle {\mathrm{\nabla }}^2}$ (or ${\displaystyle \mathrm{\Delta }}$) defined by $${\displaystyle {\mathrm{\nabla }}^{2}f=\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{y}^2}+\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{z}^2}.}$$ The Laplacian of a function is equal to the divergence of the gradient, and the trace of the Hessian matrix.