6.1.1 An Introduction to Implicit Differentiation

Question 1

An Introduction to Implicit Differentiation

Answer 1

• 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.

Question 2

note

Answer 2

• 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.

Question 3

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?

Answer 3

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

Question 4

Given y=x/4, find dydx

Answer 4

dy/dx=1/4

Question 5

Given y=4πtan3x, find dy/dx.

Answer 5

dy/dx=12πsec^23x

Question 6

Given y=3x, find dy/dx.

Answer 6

dy/dx=3

Question 7

Given y=sinx, find dy/dx.

Answer 7

dy/dx=cosx

Question 8

Given y=3x, find dx/dy.

Answer 8

dx/dy=1/3

Question 9

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

Answer 9

2

Question 10

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

Answer 10

dy/dx=2x

Question 11

Let y=1x. Find dy/dx.

Answer 11

dy/dx=−1/x^2

Question 12

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

Answer 12

dy/dx=e^x

Question 13

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

Answer 13

dy/dx=6x