Table of Contents
What are Second Order Derivatives?
A second-order derivative is a derivative of a derivative. It is a measure of how quickly a function changes with respect to changes in its first derivative. Second-order derivatives are used in mathematical models to predict how a system will behave. Second Order Derivative – Definition and Solved Examples.
Concavity
Concavity is a condition of a surface in which it curves inward. This can be seen in everyday objects such as cups, bowls, and in the surface of the Earth. The opposite of concavity is convexity, which is a condition of a surface in which it curves outward. Concavity is often associated with negative aspects such as sadness or depression, while convexity is often associated with positive aspects such as happiness or excitement.
Point of Inflection
A point of inflection (POI) is a point on a curve at which the curve changes from increasing to decreasing or from decreasing to increasing. The point of inflection is important in calculus because it is at this point that the derivative of the function is zero.
The point of inflection can be found by using the second derivative test. The second derivative is zero at the point of inflection.
For example, consider the curve y = x3. At the point of inflection, the second derivative is zero.
The point of inflection can also be found by using the equation of the curve. The equation of the curve has a turning point at the point of inflection.
For example, consider the curve y = x3 + 1. At the point of inflection, the curve has a turning point.
Second-Order Derivative Examples
1. Find the second derivative of y = x2.
The second derivative of y = x2 is 2x.
2. Find the second derivative of y = x3.
The second derivative of y = x3 is 3×2.
3. Find the second derivative of y = x.
The second derivative of y = x is 2x.
Second-Order Derivatives of a Parametric Function
In calculus, the second-order derivative of a parametric function is the derivative of the derivative of the function. This is a measure of the rate of change of the rate of change of the function, and is also called the second derivative of the function.
The second-order derivative can be used to determine whether a particular curve is a maximum or a minimum point on a graph. If the second derivative is negative at a point on the graph, then that point is a minimum; if the second derivative is positive at a point on the graph, then that point is a maximum.
