Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Calculus I
Topic: Derivatives
Subtopic: Implicit Differentiation

Overview

An implicit equation is one where there are x's and y's on both sides of the equation usually so intermingled that it is difficult to isolate the y, for example xy^3=sin(x^2y). To differentiate these equations requires a process called implicit differentiation which is the focus of today's lesson.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 2.6.1 Understand the difference between an equation written explicitly and one written implicitly
• 2.6.2 Know that the graphs of most curves given implicitly are non-functions (do not pass the vertical line test)
• 2.6.3 Implicitly differentiate to find dy/dx
• 2.6.4 Implicitly differentiate to find the second or third derivative of an implicit equation
• 2.6.5 Use mathematically correct notation and format when differentiating implicitly
• 2.6.6 Find the slope of the tangent line to a point that lies on an implicitly defined curve
• 2.6.7 Find the point(s) at which an implicitly defined curve has a horizontal or vertical tangent line

Terminology

Terms you should be able to define: implicit equation, explicit equation, implicit differentiation

Text Notes

This is an important section to get down well especially before we move into related rate applications which require implicit differentiation to solve.

Mini-Lectures and Examples

STUDY: Implicit Differentiation

Supplemental Resources (optional)

rev. 2020-10-10