Calculus BC

2014 Implicit Differentiation

Implicit Differentiation

Equation for a line:

Explicit Form

<One variable given explicitly in terms of the other>

Implicit Form

<Function implied by the equation>

  Differentiate the Explicit

< Explicit: , y is function of x >

Differentiation taking place with respect to x. The derivative is explicit also.

y mx b

Ax By C

24 3 4y x x

8 3dy xdx

Implicit Differentiation

Equation of circle:

 

To work explicitly; must work two equations

2 2 9y x

29y x

 Implicit Differentiation is a Short Cut - A method to handle equations that are not easily written explicitly.

( Usually non-functions)

29y x

Implicit Differentiation

Chain Rule Pretend y is some function like

so becomes

 (A)

(B)

(C)

Note: Use the Leibniz form. Leads to Parametric and Related Rates.

2 2 3y x x 2 4( 2 3)x x 4y

Find the derivative with respect to x

< Assuming - y is a differentiable function of x >

32y

4y

2 3x y

Implicit Differentiation

(D) Product Rule

 

 (E) Chain Rule 3( )xy

2xy

Implicit Differentiation

To find implicitly.

 

EX: Diff Both Sides of equation with respect to x

  Solve for

 

dydx

2 2 9x y dydx

EX 1:3 2 25 4y y y x

(a) Find the derivative at the point ( 5, 3 ) , at ( -1,-3 )

(b) Find where the curve has a horizontal tangent.

 (c) Find where the curve has vertical tangents.

Ex 2:

3 3 2x y xy

< Folium of Descartes >

Why Implicit?

3 3 2x y xy

< Folium of Descartes > Explicit Form:

3 6 3 3 6 33 31

1 1 1 18 82 4 2 4

y x x x x x x

3 6 3 3 6 33 32 1

1 1 1 1 13 8 82 2 4 2 4

y y x x x x x x

3 6 3 3 6 33 33 1

1 1 1 1 13 8 82 2 4 2 4

y y x x x x x x

Ex 2 Graph:

3 3 2x y xy < Folium of Descartes >

2

3 3

3 3, ; 11 1t tx y tt t

Plot the Folium of Descartes on your graphing calculator and determine the portion of the folium generated when

(a) t < -1 ; (b) -1 < t 0 ; (c) t > 0

Parametric Form:

2nd Derivatives

NOTICE:The second derivative is in terms of x , y , AND dy /dx.

The final step will be to substitute back the value of dy / dx into the second derivative.

EX: Our friendly circle. Find the 2nd Derivative.2 2 9x y

2nd DerivativesEX: Find the 2nd Derivative.

23 5xy

Higher DerivativesEX: Find the Third Derivative.

sin( )y x

Last update

• 10/19/10

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