3.7 Implicit Differentiation Niagara Falls, NY & Canada Greg Kelly, Hanford High School,...

3.7 Implicit Differentiation

Niagara Falls, NY & Canada

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2003

Before we start, we are going to load the Calculus Tools flash application software to your calculator.

1. Connect the sending and receiving calculators together.

2. On the sending unit, press , and then .

2nd VAR-LINK

2nd F7

Press to select .F4 Calculus Tools

3. On the receiving unit, press .

2nd VAR-LINK

4. On both units, press (Link).F3

5. On the receiving unit, select (Receive).

2

6. On the sending unit, select (Send). 1

2 2 1x y

This is not a function, but it would still be nice to be able to find the slope.

2 2 1d d dx y

dx dx dx Do the same thing to both sides.

2 2 0dy

x ydx

Note use of chain rule.

2 2dyy xdx

2

2

dy x

dx y

dy x

dx y

22 siny x y

22 sind d d

y x ydx dx dx

This can’t be solved for y.

2 2 cosdy dy

x ydx dx

2 cos 2dy dy

y xdx dx

22 cosdy

xydx

2

2 cos

dy x

dx y

This technique is called implicit differentiation.

1 Differentiate both sides w.r.t. x.

2 Solve for .dy

dx

We need the slope. Since we can’t solve for y, we use

implicit differentiation to solve for .dy

dx

Find the equations of the lines tangent and normal to the

curve at .2 2 7x xy y ( 1, 2)

2 2 7x xy y

2 2 0dydy

x yy xdxdx

Note product rule.

2 2 0dy dy

x y x ydx dx

22dy

y xy xdx

2

2

dy y x

dx y x

2 2 1

2 2 1m

2 2

4 1

4

5

Find the equations of the lines tangent and normal to the

curve at .2 2 7x xy y ( 1, 2)

4

5m tangent:

42 1

5y x

4 42

5 5y x

4 14

5 5y x

normal:

52 1

4y x

5 52

4 4y x

5 3

4 4y x

Higher Order Derivatives

Find if .2

2

d y

dx3 22 3 7x y

3 22 3 7x y

26 6 0x y y

26 6y y x

26

6

xy

y

2xy

y

2

2

2x y x yy

y

2

2

2x xy y

y y

2 2

2

2x xy

y

x

yy

4

3

2x xy

y y

Substitute back into the equation.

y

APPS

Select and press . Calculus Tools ENTER

If you see this

screen, press

, change the

mode settings as

necessary, and

press

again.

ENTER

APPS

Now let’s do one on the TI-89:

3 22 3 7x y Find .2

2

d y

dx

Now let’s do one on the TI-89:

3 22 3 7x y

APPS

Select and press . Calculus Tools ENTER

Press (Deriv)F2

Press (Implicit Difn)4

Enter the equation.(You may have to unlock the alpha mode.)

Set the order to 2.

Press .ENTER

Find .2

2

d y

dx

Press and then

to return your

calculator to normal.

ESC

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