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Concept
Second derivative
Summary
In calculus, the second derivative, or the second-order derivative, of a function f is the derivative of the derivative of f. 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 a is acceleration, v is velocity, t is time, x is position, and d is the instantaneous "delta" or change. The last expression is the second derivative of position (x) with respect to time.
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 is usually denoted . That is:
When using Leibniz's notation for derivatives, the second derivative of a dependent variable y with respect to an independent variable x is written
This notation is derived from the following formula:
Given the function
the derivative of f is the function
The second derivative of f is the derivative of , namely
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 will be concave up (also 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 (also simply called concave), and its tangent lines will lie above the graph of the function.
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.
Official source
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