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The flow of net fixed investment is the time derivative of the capital stock. The flow of inventory investment is the time derivative of the stock of inventories. The growth rate of the money supply is the time derivative of the money supply divided by the money supply itself. Sometimes the time derivative of a flow variable can appear in a model:
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
The higher order derivatives can be applied in physics; for example, while the first derivative of the position of a moving object with respect to time is the object's velocity, how the position changes as time advances, the second derivative is the object's acceleration, how the velocity changes as time advances.
The rate of change is usually with respect to time. Because science and engineering often relate quantities to each other, the methods of related rates have broad applications in these fields. Differentiation with respect to time or one of the other variables requires application of the chain rule, [1] since most problems involve several variables.
This states that differentiation is the reverse process to integration. Differentiation has applications in nearly all quantitative disciplines. In physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of the velocity with respect to time is acceleration.
If the input of the function represents time, then the difference quotient represents change with respect to time. For example, if f {\displaystyle f} is a function that takes a time as input and gives the position of a ball at that time as output, then the difference quotient of f {\displaystyle f} is how the position is changing in time, that ...
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as ...
D-notation leaves implicit the variable with respect to which differentiation is being done. However, this variable can also be made explicit by putting its name as a subscript: if f is a function of a variable x, this is done by writing [6] for the first derivative, for the second derivative,