隐函数求导：链式法则在显式与隐式求导中的应用对比

Hey there! I totally get where you're coming from—making the jump from explicit to implicit differentiation with the chain rule can feel like connecting two pieces that should fit but just won't click at first. Let's break this down slowly and tie what you already know about explicit functions to the implicit case.

First, Let's Recap the Chain Rule for Explicit Functions #

You already know this, but let's ground ourselves with a concrete example to set the stage. Suppose we have an explicit function like:
y = (x² + 1)³
Here, we recognize it as a composite function:

• Outer function: f(u) = u³ (where u is the inner function)
• Inner function: u(x) = x² + 1

The chain rule tells us to take the derivative of the outer function with respect to u, then multiply by the derivative of the inner function with respect to x:
dy/dx = f’(u) * u’(x) = 3u² * 2x = 3(x² + 1)² * 2x

Simple enough—we identify the "inside" and "outside" parts, differentiate each, and multiply. Now let's map this exact logic to implicit differentiation.

The Chain Rule in Implicit Differentiation: Same Logic, Hidden Inner Function #

Implicit functions are just equations where y is a function of x, but we don't write it as y = f(x) (it might even be impossible to solve for y explicitly). Let's use the classic circle equation:
x² + y² = 1

The key realization here is: y is still a function of x—we just don't have it written out. Let's rewrite the equation to make this explicit (pun intended):
x² + [f(x)]² = 1

Now, if we differentiate both sides with respect to x, we treat [f(x)]² exactly like we treated u³ in the explicit example. Let's do this step by step:

1. Differentiate x² with respect to x: that's 2x, no surprises.
2. Differentiate [f(x)]² with respect to x: this is a composite function!
   • Outer function: g(u) = u² (where u = f(x) = y)
   • Inner function: u(x) = f(x)
   • Applying the chain rule: g’(u) * u’(x) = 2u * f’(x) = 2y * dy/dx
3. Differentiate the right side (1): that's 0.

Putting it all together:
2x + 2y * dy/dx = 0
Then we just solve for dy/dx:
dy/dx = -x/y

Let's Try a Trickier Example to Solidify the Link #

Take the equation e^(xy) = x + y. Let's differentiate both sides, leaning on the chain rule every step:

1. Left side: e^(xy) is a composite function where the outer function is e^u and the inner function is u = xy.
   • Derivative of outer function with respect to u: e^u
   • Derivative of inner function u = xy with respect to x: we use the product rule here, but still apply the chain rule to y—since y is a function of x, its derivative is dy/dx. So d/dx(xy) = y + x*dy/dx.
   • Multiply them together (chain rule!): e^(xy) * (y + x*dy/dx)
2. Right side: d/dx(x + y) = 1 + dy/dx (again, y is a function of x, so its derivative is dy/dx via chain rule—think of y as f(x), so derivative of f(x) is f’(x) = dy/dx)

Now set left = right:
e^(xy)(y + x*dy/dx) = 1 + dy/dx
Then rearrange terms to solve for dy/dx:
y*e^(xy) + x*e^(xy)*dy/dx - dy/dx = 1
dy/dx(x*e^(xy) - 1) = 1 - y*e^(xy)
dy/dx = (1 - y*e^(xy))/(x*e^(xy) - 1)

The Core Connection Between Explicit and Implicit Chain Rule Use #

At the end of the day, the chain rule works the same way in both cases:

• Explicit: You have a clear inner function (like x² + 1) that's a function of x—you differentiate the outer function, multiply by the derivative of the inner function.
• Implicit: The inner function is just hidden as y (or an expression involving y), but y is still a function of x. Whenever you differentiate a term with y, you're really differentiating a composite function where y is the inner function—so you multiply by dy/dx (the derivative of the inner function with respect to x).

Quick Step-by-Step for Implicit Differentiation #

To make this actionable, here's a simple checklist:

• Differentiate both sides of the equation with respect to x.
• For any term with only x: differentiate normally (no chain rule needed here).
• For any term with y: treat y as a function of x—differentiate the "outer" part of the term, then multiply by dy/dx (this is the chain rule in action).
• Collect all terms with dy/dx on one side of the equation, move everything else to the other side.
• Factor out dy/dx and solve for it.

内容的提问来源于stack exchange，提问作者SLax