6.2.1 Using Implicit Differentiation Flashcards Preview

AP Calculus AB > 6.2.1 Using Implicit Differentiation > Flashcards

Flashcards in 6.2.1 Using Implicit Differentiation Deck (14)
Loading flashcards...
1
Q

Using Implicit Differentiation

A

• Find the derivative of a relation by differentiating each side of its equation implicitly and solving for the derivative as an unknown. This process is called implicit differentiation.

2
Q

note

A
  • Implicit differentiation often produces a derivative expressed in terms of more than one variable. When evaluating the slope of a line tangent to a point of a relation, it is necessary to substitute both the x-value and the y-value of the point into the derivative.
  • Notice that you could substitute any values for x and y into the derivative. However, only ordered pairs of the original relation produce reasonable answers.
  • Here is a complicated-looking relation.
  • Find the derivative implicitly by taking the derivative of both sides of the implicit equation.
  • Now you can differentiate each term piece by piece.
  • Sometimes you will have to use different differentiation rules in the middle of a problem. Here the product and chain rules are both used.
  • Once you have differentiated each term, you can solve
    for dy/dx.
3
Q

Given the equation 1−ln(xy)=e^y,find dydx.

A

dy/dx=−y/xye^y+x

4
Q

Given the equation xy=5,find dy/dx.

A

dy/dx=−y/x

5
Q

Given the equation sinxy=1/2,find dy/dx.

A

dy/dx=−y/x

6
Q

Given the equation x^2y+y^2x=0,find dy/dx.

A

dy/dx=−2xy−y^2/2xy+x^2

7
Q

Given x/y=1/9,find dy/dx.

A

dy/dx=9

8
Q

Suppose a curve is defined by the equation (6−x)y^2=x^3. What is the equation of the line tangent to the curve at (3, 3)?

A

y=2x−3

9
Q

Given the equation2x+2y+xy^2=5,find dy/dx.

A

dy/dx=−2+y^−2/2−2xy^−3

10
Q

Suppose a curve is defined by the equation y^2=x^3(2−x). What is the equation of the line tangent to the curve at (1,−1)?

A

y=−x

11
Q

Given the equation x^2+3x=y^2+y−6,find dy/dx.

A

dy/dx=2x+3/2y+1

12
Q

Given cos^3(sinxy)=k where k is some constant,find dxdy

A

dx/dy=−x/y

13
Q

Given the equation xy=cotxy,find dy/dx.

A

dy/dx=−y/x

14
Q

Given sin^3(cosxy)=k where k is some constant, find dy/dx.

A

dy/dx=−y/x

Decks in AP Calculus AB Class (190):