The Derivative: Calculating Instantaneous Rates Of Change In Math

derivative of x with respect to time

instantaneous velocity (can be positive, negative, or 0)

The derivative of x with respect to time is commonly denoted by the symbol dx/dt. It represents the rate of change of x with respect to time t.

Mathematically, if x(t) is a function of time, the derivative dx/dt can be calculated using the limit definition of the derivative as:

dx/dt = lim (h->0) [x(t+h) – x(t)] / h

This expression represents the instantaneous rate of change of x at time t. In other words, it gives the slope of the tangent line to the curve of x(t) at time t.

For example, if x(t) represents the position of an object at time t, then dx/dt represents the object’s velocity at time t. Similarly, if x(t) represents the temperature of a system at time t, then dx/dt represents the rate of change of temperature with respect to time (i.e. how fast the temperature is changing).

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