Flashcards in 6.1.1 An Introduction to Implicit Differentiation Deck (13):
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.
- 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
Given y=4πtan3x, find dy/dx.
Given y=3x, find dy/dx.
Given y=sinx, find dy/dx.
Given y=3x, find dx/dy.
Suppose a curve is defined by the equation x^2+y^2=4.How many lines are tangent to the curve where x=0?
Find dy/dx, where y=x^2.
Let y=1x. Find dy/dx.
Given y=e^x, find dy/dx.