A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as $t$.

A variety of notations are used to denote the time derivative. In addition to the normal notation,

$\backslash frac\; \{dx\}\; \{dt\}$

two very common shorthand notations are also used: adding a dot over the variable, $\backslash dot\{x\}$, and adding a prime to the variable, $x\text{'}$. These two shorthands are generally not mixed in the same set of equations.

Higher time derivatives are also used: the second derivative with respect to time is written as

$\backslash frac\; \{d^2x\}\; \{dt^2\}$

with the corresponding shorthands of $\backslash ddot\{x\}$ and $x\text{'}\text{'}$.

Time derivatives are a key concept in physics. For example, for a changing position $x$, its time derivative $\backslash dot\{x\}$ is its velocity, and its second derivative with respect to time, $\backslash ddot\{x\}$, is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk.

A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another:

• force is the time derivative of momentum
• power is the time derivative of energy
• electrical current is the time derivative of electric charge

See also

• Differential calculus

Differential calculus

时域微分