In calculus, the second derivative, or the second-order derivative, of a function is the derivative of the derivative of . Informally, the second derivative can be phrased as "the rate of change of the rate of change"; for example, the second derivative of the position of an object with respect to time is the instantaneous acceleration of the object, or the rate at which the velocity of the object is changing with respect to time. In Leibniz notation:where is acceleration, is velocity, is time, is position, and d is the instantaneous "delta" or change. The last expression
\tfrac{d2x}{dt2}
On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the opposite way.
The power rule for the first derivative, if applied twice, will produce the second derivative power rule as follows:
The second derivative of a function
f(x)
f''(x)
Given the functionthe derivative of is the functionThe second derivative of is the derivative of
f'
The second derivative of a function can be used to determine the concavity of the graph of . 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.
See 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.
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
f''(x)<0
f
x
f''(x)>0
f
x
f''(x)=0
x
It is possible to write a single limit for the second derivative:
The limit is called the second symmetric derivative.[3] [4] 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: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
f
sgn(x)
The sign function is not continuous at zero, and therefore the second derivative for
x=0
x=0
Just as the first derivative is related to linear approximations, the second derivative is related to the best quadratic approximation for a function . This is the quadratic function whose first and second derivatives are the same as those of at a given point. The formula for the best quadratic approximation to a function around the point isThis quadratic approximation is the second-order Taylor polynomial for the function centered at .
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]
v(0)=v(L)=0
λj=-\tfrac{j2\pi2}{L2}
For other well-known cases, see Eigenvalues and eigenvectors of the second derivative.
See main article: Hessian matrix. The second derivative generalizes to higher dimensions through the notion of second partial derivatives. For a function, these include the three second-order partialsand the mixed partials
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.)
See main article: Laplace operator. Another common generalization of the second derivative is the Laplacian. This is the differential operator
\nabla2
\Delta