偏导数是多元微积分中的基本概念。与处理单变量函数的普通导数不同，偏导数帮助我们理解多元函数如何变化。当我们有一个多变量函数时，偏导数测量相对于一个变量的变化率，同时将所有其他变量视为常数。记号使用偏导符号∂而不是普通导数符号d。
偏导数的严格定义使用极限概念。对于函数f关于x的偏导数，我们计算当x变化一个小量h时函数值的变化率，同时保持y不变。类似地，对于关于y的偏导数，我们保持x不变而让y变化。在实际计算中，我们将其他变量视为常数，只对目标变量求导。
从几何角度看，偏导数具有清晰的意义。对于二元函数的图像（一个三维曲面），关于x的偏导数表示在固定y值时，曲面在x方向上的斜率。这相当于用一个垂直于x轴的平面切割曲面，得到的曲线在该点的切线斜率。类似地，关于y的偏导数给出了在y方向上的斜率。
让我们看一些具体的计算例子。对于多项式函数，我们按照幂函数的求导规则，将其他变量视为常数。对于三角函数和指数函数，我们应用链式法则。偏导数在许多领域都有重要应用，包括优化问题、物理学中的梯度概念、经济学中的边际分析，以及现代机器学习中的梯度下降算法。
Now let's explore the geometric interpretation of partial derivatives. When we have a 3D surface representing a function z equals f of x y, partial derivatives have a clear geometric meaning. The partial derivative with respect to x represents the slope of the curve we get when we slice the surface with a vertical plane where y is constant. Similarly, the partial derivative with respect to y gives us the slope when x is held constant. These cross-sections reveal how the surface changes in each direction.
Let's examine the formal mathematical definition of partial derivatives using limits. The partial derivative of f with respect to x is defined as the limit of the difference quotient as h approaches zero, while keeping y fixed. Similarly for the partial derivative with respect to y. Let's work through a concrete example with f of x y equals x squared y plus 3 x y. To find the partial derivative with respect to x, we treat y as a constant and differentiate, giving us 2xy plus 3y, which factors as y times 2x plus 3. For the partial derivative with respect to y, we treat x as constant, yielding x squared plus 3x, or x times x plus 3.
Now let's establish practical methods for computing partial derivatives using familiar differentiation rules. The key principle is to treat other variables as constants and apply standard rules like the power rule, product rule, and chain rule. Let's work through examples of increasing complexity. For polynomial functions like 3x squared y plus 4xy cubed minus 2y squared, we differentiate term by term. The partial with respect to x gives 6xy plus 4y cubed, treating y as constant. For trigonometric functions involving products like sine of xy plus cosine of x squared, we use the chain rule. For exponential functions like e to the xy plus x squared e to the y, we apply both the chain rule and product rule as needed.
Partial derivatives demonstrate remarkable real-world relevance across multiple fields. In optimization problems, we find critical points by setting both partial derivatives equal to zero and solving the resulting system. Let's examine a concrete example with f of x y equals x squared plus y squared minus 4x minus 6y plus 13. First, we compute the partial derivatives: partial f partial x equals 2x minus 4, and partial f partial y equals 2y minus 6. Setting both equal to zero gives us the system 2x minus 4 equals zero and 2y minus 6 equals zero. Solving yields x equals 2 and y equals 3, so our critical point is at coordinates 2, 3. Beyond optimization, partial derivatives appear in heat distribution analysis, economic profit maximization, gradient descent algorithms in artificial intelligence, and engineering optimization problems.