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AP Calculus AB: 6.1.1 An Introduction to Implicit Differentiation

Mathematics13 CardsCreated 3 months ago

This flashcard set introduces the concept of implicit differentiation, emphasizing the difference between functions and general relations. It explains how to use Leibniz notation to take derivatives of relations that aren't functions and includes foundational examples to build comfort with differentiation techniques applied beyond standard function forms.

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An Introduction to Implicit Differentiation

  • The definition of the derivative empowers you to take derivatives of functions, not relations.

  • Leibniz notation is another way of writing derivatives. This notation will be helpful when finding derivatives of relations that are not functions.

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Key Terms

Term
Definition

An Introduction to Implicit Differentiation

  • The definition of the derivative empowers you to take derivatives of functions, not relations.

  • Leibniz notation is another w...

note

  • A function is a set of ordered pairs in which each domain value is mapped to at most one range value.

  • A relation is a set of...

Suppose a curve is defined by the equation(x−2)^2/4−(y+2)^2/9=1.Is this curve a function? Why or why not?

No, the curve is not a function because the curve does not pass the vertical line test.

Given y=x/4, find dydx

dy/dx=1/4

Given y=4πtan3x, find dy/dx.

dy/dx=12πsec^23x

Given y=3x, find dy/dx.

dy/dx=3

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AP Calculus AB: 6.1.1 An Introduction to Implicit Differentiation
TermDefinition

An Introduction to Implicit Differentiation

  • The definition of the derivative empowers you to take derivatives of functions, not relations.

  • Leibniz notation is another way of writing derivatives. This notation will be helpful when finding derivatives of relations that are not functions.

note

  • A function is a set of ordered pairs in which each domain value is mapped to at most one range value.

  • A relation is a set of ordered pairs. Relations can map a
    single domain value to many range values.

  • Notice that a function counts as a relation, but a relation is not necessarily a function. A function is a special type of relation.

  • You can check to see if a graph represents a relation or a function by using the vertical line test.

  • You have not learned how to take the derivative of a relation. But notice that relations should have derivatives, since a relation can have a tangent line.

  • The circle is a common example of a relation. Notice that the circle fails the vertical line test. The circle still has tangents, however.

  • You can use Leibniz notation to make finding the derivative of a relation easier.

  • To use Leibniz notation, simply take the derivative of each side of the equation separately.

  • Notice that the derivative of y with respect to x has a special name in Leibniz notation: dy/dx.

  • Take the derivative of the right side of the equation piece by piece.

  • This piece-by-piece approach will also work when finding the derivative of a relation.

Suppose a curve is defined by the equation(x−2)^2/4−(y+2)^2/9=1.Is this curve a function? Why or why not?

No, the curve is not a function because the curve does not pass the vertical line test.

Given y=x/4, find dydx

dy/dx=1/4

Given y=4πtan3x, find dy/dx.

dy/dx=12πsec^23x

Given y=3x, find dy/dx.

dy/dx=3

Given y=sinx, find dy/dx.

dy/dx=cosx

Given y=3x, find dx/dy.

dx/dy=1/3

Suppose a curve is defined by the equation x^2+y^2=4.How many lines are tangent to the curve where x=0?

2

Find dy/dx, where y=x^2.

dy/dx=2x

Let y=1x. Find dy/dx.

dy/dx=−1/x^2

Given y=e^x, find dy/dx.

dy/dx=e^x

Given y=3x^2, find dy/dx.

dy/dx=6x