Second derivative

The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:

$${\displaystyle {\frac {d^{2}}{dx^{2}}}x^{n}={\frac {d}{dx}}{\frac {d}{dx}}x^{n}={\frac {d}{dx}}\left(nx^{n-1}\right)=n{\frac {d}{dx}}x^{n-1}=n(n-1)x^{n-2}.}$$

The second derivative of a function ${\displaystyle f(x)}$ is usually denoted ${\displaystyle f''(x)}$.[1][2] That is:

$${\displaystyle f''=\left(f'\right)'}$$

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

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

This notation is derived from the following formula:

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

Given the function

$${\displaystyle f(x)=x^{3},}$$

the derivative of f is the function

$${\displaystyle f'(x)=3x^{2}.}$$

The second derivative of f is the derivative of ${\displaystyle f'}$, namely

$${\displaystyle f''(x)=6x.}$$

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

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

The limit is called the second symmetric derivative.[3][4] Note that 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:

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

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 ${\displaystyle \operatorname {sgn}(x)}$, which is defined as:

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

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

$${\displaystyle {\begin{aligned}\lim _{h\to 0}{\frac {\operatorname {sgn}(0+h)-2\operatorname {sgn}(0)+\operatorname {sgn}(0-h)}{h^{2}}}&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)-2\cdot 0+\operatorname {sgn}(-h)}{h^{2}}}\\&=\lim _{h\to 0}{\frac {\operatorname {sgn}(h)+(-\operatorname {sgn}(h))}{h^{2}}}=\lim _{h\to 0}{\frac {0}{h^{2}}}=0.\end{aligned}}}$$

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

$${\displaystyle f(x)\approx f(a)+f'(a)(x-a)+{\tfrac {1}{2}}f''(a)(x-a)^{2}.}$$

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

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 ${\displaystyle \lambda _{j}=-{\tfrac {j^{2}\pi ^{2}}{L^{2}}}}$ and the corresponding eigenvectors (also called eigenfunctions) are ${\displaystyle v_{j}(x)={\sqrt {\tfrac {2}{L}}}\sin \left({\tfrac {j\pi x}{L}}\right)}$. Here, ${\displaystyle v''_{j}(x)=\lambda _{j}v_{j}(x)}$, for ${\displaystyle j=1,\ldots ,\infty }$.

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

• Chirpyness, second derivative of instantaneous phase
• Finite difference, used to approximate second derivative
• Second partial derivative test
• Symmetry of second derivatives

• Discrete Second Derivative from Unevenly Spaced Points