Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is...

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Section 2.6 Implicit Differentiation

Transcript of Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is...

Page 1: Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is a explicit function of x if the formula is written.

Section 2.6

Implicit Differentiation

Page 2: Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is a explicit function of x if the formula is written.

EXPLICIT AND IMPLICIT FUNCTIONS

Definition: The function y is a explicit function of x if the formula is written in the form

y = expression with x’s.

If the formula is NOT written this way, we say that y is an implicit function of x.

Page 3: Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is a explicit function of x if the formula is written.

EXAMPLE

Below is the same function defined explicitly and implicitly.

EXPLICIT: y = x2/3

IMPLICIT: y3 = x2

Page 4: Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is a explicit function of x if the formula is written.

IMPLICIT DIFFERENTIATION

Many times if a function is defined implicitly, it is difficult (or impossible) to write it in explicitly. To take the derivative of an implicitly defined function, we use implicit differentiation.

Page 5: Section 2.6 Implicit Differentiation. EXPLICIT AND IMPLICIT FUNCTIONS Definition: The function y is a explicit function of x if the formula is written.

PROCEDURE FOR IMPLICIT DIFFERENTIATION

1. Assuming y is a function of x, differentiate both sides with respect to x using the chain rule for y terms.

2. Collect the terms with dy/dx on one side and the other terms on the other side.

3. Factor out dy/dx.

4. Divide both sides by the factor to leave dy/dx by itself.


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