AQA Mathematics: Partial Differentiation Key Points | AQA 数学：偏微分 考点精讲

📚 AQA Mathematics: Partial Differentiation Key Points | AQA 数学：偏微分 考点精讲

Partial differentiation is a cornerstone of AQA Further Mathematics, extending single-variable calculus to functions of two or more variables. It finds extensive use in modelling rates of change, optimisation, and understanding geometric surfaces. Mastering this topic is essential for tackling advanced problems in pure mathematics and applied contexts.

偏微分是 AQA 进阶数学的基石，它将单变量微积分扩展到两个或更多变量的函数。偏微分广泛用于变化率建模、优化问题以及几何曲面的理解。掌握这一主题对于解决纯数学和应用背景中的高级问题至关重要。

1. What is Partial Differentiation? | 什么是偏微分？

Partial differentiation deals with functions of several variables, such as f(x, y) or f(x, y, z). Unlike ordinary differentiation, where there is only one independent variable, here we find the rate of change of the function with respect to one variable while holding the others constant.

偏微分处理的是多变量函数，例如 f(x, y) 或 f(x, y, z)。与只有一个自变量的普通微分不同，这里我们求函数关于某一个变量的变化率，同时将其他变量视为常数。

For instance, given f(x, y) = x²y + 3xy³, we can differentiate with respect to x treating y as constant, obtaining the partial derivative ∂f/∂x.

例如，给定 f(x, y) = x²y + 3xy³，我们可以对 x 求偏导数，将 y 视为常数，得到 ∂f/∂x。

The partial derivative ∂f/∂x at a point (a, b) gives the slope of the tangent line to the surface z = f(x, y) in the direction of the x-axis, keeping y fixed at b.

在点 (a, b) 处的偏导数 ∂f/∂x 给出了曲面 z = f(x, y) 在 y 固定为 b 的情况下沿 x 轴方向的切线斜率。

2. First-Order Partial Derivatives | 一阶偏导数

To compute ∂f/∂x, differentiate f with respect to x, treating all other variables (like y, z) as constants. Similarly, ∂f/∂y is found by differentiating with respect to y, holding x and any other variables constant.

计算 ∂f/∂x 时，对 x 求导，将其他所有变量（如 y, z）视为常数。同样，求 ∂f/∂y 时对 y 求导，将 x 及其他变量视为常数。

Example: Let f(x, y) = sin(xy) + e^(x+y). Then ∂f/∂x = y cos(xy) + e^(x+y), and ∂f/∂y = x cos(xy) + e^(x+y).

示例：设 f(x, y) = sin(xy) + e^(x+y)，则 ∂f/∂x = y cos(xy) + e^(x+y)，∂f/∂y = x cos(xy) + e^(x+y)。

Partial derivatives can be evaluated at specific points to give numerical values. For instance, at (1, 0), ∂f/∂x = 0·cos(0) + e¹ = e.

偏导数可以在特定点处求值，得到具体的数值。例如，在 (1, 0) 处，∂f/∂x = 0·cos(0) + e¹ = e。

3. Notation and Calculation | 符号与计算

Common notations for first-order partial derivatives include ∂f/∂x, ∂f/∂y, and ∂z/∂x if z = f(x, y). The curly ‘d’ symbol ∂ distinguishes partial derivatives from ordinary ones.

常用的一阶偏导数符号包括 ∂f/∂x、∂f/∂y，以及当 z = f(x,y) 时的 ∂z/∂x。花写的 “d” 符号 ∂ 用于区分偏导数与普通导数。

When calculating, remember that functions like ln(xy) require the chain rule just as in ordinary differentiation, but with respect to one variable. For f(x

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