对比偏导数与单变量导数的极限定义，求解单变量导数理解困惑

Hey there, let’s work through this together—you’ve already got a solid grasp of the single-variable derivative’s core definition, so let’s first untangle your confusion about the limit, then break down how partial derivatives fit in.

First: Why the Derivative Doesn’t Always Go to Infinity as ( h \to 0 ) #

Your intuition about the denominator approaching 0 makes sense at first glance, but you’re missing a key piece: the numerator is also approaching 0 (for any continuous function, which is a prerequisite for differentiability). The derivative isn’t just "denominator goes to 0"—it’s the ratio of how fast the numerator and denominator shrink to 0.

Let’s use a concrete example to make this clear:
Take ( f(x) = x^2 ). The numerator in the derivative definition is:
$$f(x+h) - f(x) = (x+h)^2 - x^2 = 2xh + h^2$$
When we divide by ( h ) (for ( h \neq 0 )), we get:
$$\frac{2xh + h^2}{h} = 2x + h$$
Now as ( h \to 0 ), this simplifies to ( 2x )—a finite value, not infinity. The numerator shrinks at a rate proportional to ( h ), matching the denominator’s shrinkage, so their ratio stays finite.

Only in specific cases where the numerator shrinks slower than the denominator does the derivative tend to infinity. For example, ( f(x) = \sqrt{x} ) at ( x=0 ):
$$\frac{\sqrt{0+h} - \sqrt{0}}{h} = \frac{\sqrt{h}}{h} = \frac{1}{\sqrt{h}}$$
As ( h \to 0^+ ), this does go to positive infinity—but this is a special case where the function isn’t differentiable at that point, not the norm.

Single-Variable Derivative vs Partial Derivative: Limit Definitions Compared #

Let’s formalize the two definitions side by side to highlight their similarities and differences.

1. Single-Variable Derivative #

For a function ( f(x) ) with one independent variable, the derivative at point ( x ) is:
$$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$

• What it means: This measures the instantaneous rate of change of ( f(x) ) as we vary the only variable ( x ) around the point ( x ). There’s only one direction to "move" (along the x-axis), so the derivative captures the function’s slope at that point.

2. Partial Derivative (for Multivariable Functions) #

Take a two-variable function ( f(x,y) ). The partial derivative with respect to ( x ) at point ( (x,y) ) is:
$$\frac{\partial f}{\partial x}(x,y) = \lim_{h\to 0}\frac{f(x+h,y)-f(x,y)}{h}$$
Similarly, the partial derivative with respect to ( y ) is:
$$\frac{\partial f}{\partial y}(x,y) = \lim_{h\to 0}\frac{f(x,y+h)-f(x,y)}{h}$$

• Key differences from single-variable derivatives:
  • We hold all other variables constant (e.g., ( y ) stays fixed when taking ( \frac{\partial f}{\partial x} )).
  • Instead of measuring the rate of change along the only possible axis, partial derivatives measure the rate of change along one specific axis (x or y) while keeping other inputs fixed.
  • Conceptually, a partial derivative is just a single-variable derivative in disguise: if you fix ( y = y_0 ), ( f(x,y_0) ) becomes a function of ( x ) alone, and ( \frac{\partial f}{\partial x}(x,y_0) ) is exactly its single-variable derivative at ( x ).

Recap #

• The derivative’s limit isn’t about the denominator approaching 0 in isolation—it’s the ratio of two shrinking quantities, which can be finite, infinite, or undefined depending on how fast each shrinks.
• Partial derivatives extend the single-variable derivative idea to multivariable functions by focusing on one variable at a time, holding others constant.

内容的提问来源于stack exchange，提问作者sangstar