Implicit Functions

Definition

Definition of Implicit Functions

A functional relationship determined by the equation $F(x, y) = 0$ is called an implicit function.

Characteristics

Implicit functions have the following characteristics:

Non-explicit: Cannot directly solve for an explicit expression of y in terms of x
Equation-determined: The functional relationship needs to be determined through the equation
Multi-valuedness: Multiple function values may correspond to the same independent variable value
Locality: Implicit functions are usually valid only in certain regions

Common Examples

Circle Equation

$x^2 + y^2 = 1$ determines an implicit function, which can be solved as:

$y = \sqrt{1 - x^2}$ (upper semicircle)
$y = -\sqrt{1 - x^2}$ (lower semicircle)

Ellipse Equation

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ determines an implicit function

Other Examples

$x^3 + y^3 = 3xy$ (Folium of Descartes)
$e^x + e^y = 1$
$\sin(x + y) = x + y$

Implicit Function Differentiation Methods

Basic Steps

Differentiate both sides of the equation with respect to x
Use the chain rule to handle the derivative of y
Solve for $\frac{dy}{dx}$

Specific Method

For the equation $F(x, y) = 0$ :

Differentiate both sides with respect to x: $\frac{\partial F}{\partial x} + \frac{\partial F}{\partial y} \cdot \frac{dy}{dx} = 0$
Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = -\frac{\frac{\partial F}{\partial x}}{\frac{\partial F}{\partial y}}$

Notes

Denominator not zero: $\frac{\partial F}{\partial y} \neq 0$
Locality: The result is valid only near certain points
Multi-valuedness: May need to consider multiple branches

Examples of Implicit Function Differentiation

Example 1: Derivative of Circle

For $x^2 + y^2 = 1$ :

Differentiate both sides: $2x + 2y \cdot \frac{dy}{dx} = 0$
Solve for the derivative: $\frac{dy}{dx} = -\frac{x}{y}$

Example 2: Derivative of Ellipse

For $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ :

Differentiate both sides: $\frac{2x}{a^2} + \frac{2y}{b^2} \cdot \frac{dy}{dx} = 0$
Solve for the derivative: $\frac{dy}{dx} = -\frac{b^2 x}{a^2 y}$

Exercises

Exercise 1

Find the derivative of the implicit function $x^2 + y^2 = 4$ at the point $(1, \sqrt{3})$ .

Reference Answer (2 个标签)

implicit function differentiation

Solution Approach: Differentiate both sides of the equation with respect to x, then solve for $\frac{dy}{dx}$ .

Detailed Steps:

Differentiate both sides of $x^2 + y^2 = 4$ with respect to x: $2x + 2y \cdot \frac{dy}{dx} = 0$
Solve for $\frac{dy}{dx}$ : $2y \cdot \frac{dy}{dx} = -2x$ $\frac{dy}{dx} = -\frac{x}{y}$
Substitute the point $(1, \sqrt{3})$ : $\frac{dy}{dx} = -\frac{1}{\sqrt{3}} = -\frac{\sqrt{3}}{3}$

Answer: The derivative at point $(1, \sqrt{3})$ is $-\frac{\sqrt{3}}{3}$ .

Exercise 2

Find the derivative of the implicit function $x^3 + y^3 = 3xy$ .

Reference Answer (2 个标签)

implicit function differentiation

Solution Approach: Differentiate both sides of the equation with respect to x, then solve for $\frac{dy}{dx}$ .

Detailed Steps:

Differentiate both sides of $x^3 + y^3 = 3xy$ with respect to x: $3x^2 + 3y^2 \cdot \frac{dy}{dx} = 3y + 3x \cdot \frac{dy}{dx}$
Rearrange the equation: $3y^2 \cdot \frac{dy}{dx} - 3x \cdot \frac{dy}{dx} = 3y - 3x^2$ $(3y^2 - 3x) \cdot \frac{dy}{dx} = 3y - 3x^2$
Solve for $\frac{dy}{dx}$ : $\frac{dy}{dx} = \frac{3y - 3x^2}{3y^2 - 3x} = \frac{y - x^2}{y^2 - x}$

Answer: $\frac{dy}{dx} = \frac{y - x^2}{y^2 - x}$ .

Exercise 3

Find the derivative of the implicit function $e^x + e^y = 1$ .

Reference Answer (2 个标签)

implicit function differentiation

Solution Approach: Differentiate both sides of the equation with respect to x, then solve for $\frac{dy}{dx}$ .

Detailed Steps:

Differentiate both sides of $e^x + e^y = 1$ with respect to x: $e^x + e^y \cdot \frac{dy}{dx} = 0$
Solve for $\frac{dy}{dx}$ : $e^y \cdot \frac{dy}{dx} = -e^x$ $\frac{dy}{dx} = -\frac{e^x}{e^y} = -e^{x-y}$

Answer: $\frac{dy}{dx} = -e^{x-y}$ .

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$F(x, y)$	Mathematical Symbol	F of x and y	Function of two variables
$F(x, y) = 0$	Mathematical Symbol	F of x and y equals zero	Implicit function equation
$\frac{dy}{dx}$	Mathematical Symbol	dy by dx	Derivative of y with respect to x
$\frac{\partial F}{\partial x}$	Mathematical Symbol	partial F by partial x	Partial derivative of $F$ with respect to $x$
$\frac{\partial F}{\partial y}$	Mathematical Symbol	partial F by partial y	Partial derivative of $F$ with respect to $y$
$e^x$	Mathematical Symbol	e to the power x	Exponential function with base $e$
$e^y$	Mathematical Symbol	e to the power y	Exponential function with base $e$
$\sin(x + y)$	Mathematical Symbol	sine of x plus y	Sine function

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
隐函数	implicit function	/ɪmˈplɪsɪt ˈfʌŋkʃən/	Functional relationship determined by an equation
显式函数	explicit function	/ɪkˈsplɪsɪt ˈfʌŋkʃən/	Function that can be directly expressed as y in terms of x
隐函数求导	implicit differentiation	/ɪmˈplɪsɪt ˌdɪfəˌrenʃɪˈeɪʃən/	Method for finding derivatives of implicit functions
偏导数	partial derivative	/ˈpɑːʃəl dɪˈrɪvətɪv/	Derivative of a multivariable function with respect to one variable
链式法则	chain rule	/tʃeɪn ruːl/	Rule for differentiating composite functions
多值性	multi-valuedness	/ˌmʌlti ˈvæljudnəs/	Property where one independent variable corresponds to multiple function values
局部性	locality	/ləʊˈkælɪti/	Property where function is valid only in certain regions
分支	branch	/brɑːntʃ/	Different solutions of implicit functions
笛卡尔叶形线	Folium of Descartes	/ˈfəʊliəm əv deɪˈkɑːrt/	Curve determined by the equation $x^3 + y^3 = 3xy$

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