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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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