Do NowWhich of the following statements could be true if ?

13( )f x x

739( ) 2

28f x x

433( ) 6

4f x x

A) B)

C) D)433( ) 4

4f x x

739( ) 9

28f x x

Only A and C could be true(but not simultaneously…)

Section 3.7b

Implicit Differentiati

on

We can use Imp. Diff. with other familiar processes,including finding tangents, normals, and/or higher-order derivatives…

(when )

3 22 3 8d dx ydx dx

26 6 0x yy 2 0x yy

2yy x

2xyy

0y

Find the second derivative: 3 22 3 8x y

To find the second derivative, simply differentiate again:

…and substitute for y’ so that we have y’’ in terms of x and y:

2d xydx y

2

2

2 1y x x yy

2

2

2x x yy y

2 2

2

2x x xyy y y

4

3

2x xy y

(when )0y

Practice ProblemsFind the second derivative: 3 3 1x y

2 23 3 0dyx ydx

2

2

dy xdx y

2 22

22 2

2 2 dyy x x yd y dxdx y

22 2

2

4

2 2 xxy x yy

y

Practice ProblemsFind the second derivative: 3 3 1x y

22 2

2

4

2 2 xxy x yy

y

42

4

22 xxyy

y

3 4

4

2 2 1xy xy y

3 4

5

2 2xy xy

3 3

5

2x y x

y

5

2xy

Practice ProblemsFind the second derivative: 3 2cosy y x

23 2sindy dyy xdx dx

23 1 2sindy y xdx

2

2sin3 1

dy xdx y

22

22 2

3 1 2cos 2sin 6

3 1

dyy x x yd y dxdx y

Practice ProblemsFind the second derivative: 3 2cosy y x

22

22 2

3 1 2cos 2sin 6

3 1

dyy x x yd y dxdx y

22

22

2sin2cos 3 1 12 sin3 1

3 1

xx y y xy

y

2 2

22 2

2cos 3 1 24 sin 13 1 3 1

x y y x

y y

Practice ProblemsFind the second derivative: 3 2cosy y x

2 2

22 2

2cos 3 1 24 sin 13 1 3 1

x y y x

y y

2 2

32

2cos 3 1 24 sin

3 1

x y y x

y

2 2

32

cos 3 1 12 sin2

3 1

x y y x

y

Practice ProblemsFind the tangent and normal to the ellipseat the point (–1, 2).

2 2 7x xy y

First, find slope:

Next, evaluate slope at (–1, 2):

2 2 0dy dyx x y ydx dx

2 2dyy x y xdx

2

2dy y xdx y x

1,2

dydx

2 2 12 2 1

45

Practice ProblemsFind the tangent and normal to the ellipseat the point (–1, 2).

2 2 7x xy y

Point: 1,2 Slope:45

Tangent

42 15

y x 4 145 5

y x

Normal

52 14

y x 5 34 4

y x

Practice ProblemsFor each of the following, find the lines that are (a) tangent and(b) normal to the given curve at the given point.

Tangent: Normal:

2,1

3y x 12

dydx y

1y x

2 2 4 1 0y x y

Practice ProblemsFor each of the following, find the lines that are (a) tangent and(b) normal to the given curve at the given point.

Tangent: Normal:

4, 2

1 52 8

y x

2 sin 2 sin 22 cos 2 cos 2

dy y x ydx x y x

2y x

sin 2 cos 2x y y x

Practice ProblemsFor each of the following, find the lines that are (a) tangent and(b) normal to the given curve at the given point.

Tangent: Normal:

0,

0x 2

2 cos1 2 sin

dy x ydx x y

πy

2 2cos sin 0x y y