Implicit Differentiation
Copyright Maplesoft, a division of Waterloo Maple Inc., 2007
Introduction
This application is one of a collection of examples teaching Calculus with Maple. These applications use Clickable Calculus? methods to solve problems interactively. Steps are given at every stage of the solution, and many are illustrated using short video clips. Click on the 
buttons to watch the videos.
The steps in the document can be repeated to solve similar problems.
Problem Statement
The equation 
defines 
implicitly. Obtain the derivative 
by the technique of implicit differentiation.
Solution
|
Enter the given equation:
|
|
|
 |
(3.1) |
|
Obtain a graph of the implicitly defined function in  .
|
Right-click, Plots>Plot Builder. For the x range, choose 0 to 1 and for the y range, choose -1 to 1. Click on Options button, change grid size to 100x100. Click Plot button.
|
|
|
Rewrite the equation to make the independent variable explicit:
|
Copy the given equation to a new line or use [Ctrl][L] and enter the equation label. Press [Enter]. Right-click, Evaluate at a point.
Set y = y(x).
|
|
 |
(3.2) |
 |
(3.3) |
|
Differentiate with respect to the independent variable:
|
Right-click, Differentiate>x.
|
|
 |
(3.4) |
|
Isolate the derivative:
|
Right-click, Solve>Isolate Expression for>diff(y(x),x)
|
|
 |
(3.5) |
|
Make the independent variable implicit:
|
Copy expression, paste on new line, edit the right-hand side to change all y(x) to y.
|
|
 |
(3.6) |
Tutors and Templates
Maple has an implicit differention task Template, which can be used to step through problems easily. To create a template in your own document, select Tools > Tasks > Browse, and then navigate to Calculus / Derivatives / Implicit Differentiation.
Implicit Differentiation with Two Variables
Using implicit differentiation ., compute the derivative 
for the function 
defined implicitly by the equation 
|
Implicit Differentiation
|
|
Enter an equation in two variables:
|
|
|
Dependent variable:  Independent variable: 
|
|
Implicit Derivative:
|
|
|
Stepwise Calculation
|
|
Make dependent variable explicit:
|
|
|
Differentiate with respect to independent variable:
 
|
|
|
Isolate Derivative:
|
|
|
Make independent variable implicit:
|
|
|
|
Related Tutors
Tools > Tutors > Calculus - Single Variable > Derivatives
Tools > Tutors > Calculus - Single Variable > Differentiation Methods
Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended to demonstrate the use of Maple to solve a particular problem. It has been made available for product evaluation purposes only and may not be used in any other context without the express permission of Maplesoft.
