The Derivative
导数
| Table:derivatives of functions
函数的导数表 |
Derivative concepts
索引导数概念
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The Derivative导数
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The Partial Derivative偏导数
The ordinary derivative of a function of one variable can be carried out because everything else in the function is a constant and does not affect the process of differentiation. When there is more than one variable in a function it is often useful to examine the variation of the function with respect to one of the variables with all the other variables constrained to stay constant. This is the purpose of a partial derivative. The derivative is carried out in the same way as ordinary differentiation with this constraint. For example, given the polynomial in variables x and y, 单变量函数的普通导数可以计算，因为函数中其他部分都是常量，不会影响求导过程。当函数中包含多个变量时，通常有用处的是考察函数相对于其中一个变量的变化，同时将其他变量保持不变。这就是偏导数的目的。偏导数的计算方式与普通求导相同，只不过要满足这个约束条件。例如，给定一个包含x和y两个变量的多项式，
 the partial derivative with respect to x is written 对x的偏导数写作
 and the partial derivative with respect to y is written 并且关于y的偏导数写作
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Time Derivatives时间导数
Time derivatives are the standard way of representing instantaneous velocities and accelerations. 时间导数是表示瞬时速度和加速度的标准方法。
 One example of the use of the derivative is in obtaining the velocity and acceleration from a position equation. 导数的一个例子是通过位置方程求得速度和加速度。
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Derivatives of Polynomials多项式导数
Many functions in physical problems have the form of polynomials. The derivative of a polynomial is the sum of the derivatives of its terms, and for a general term of a polynomial such as 许多物理问题中的函数具有多项式形式。多项式的导数是其各项导数之和，而对于一般的多项式项如
 the derivative is given by 导数由
 One of the common applications of this is in the time derivatives leading to the constant acceleration motion equations. 这一应用之一是时间导数导致恒定加速度运动方程的产生。
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