3.7 Implicit
Differentiation
Niagara Falls, NY & Canada
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
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
F7 .
then 2nd
Press
F4
2nd
VAR-LINK
, and
to select Calculus Tools .
3. On the receiving unit, press
4. On both units, press
F3
4. On the receiving unit, select
5. On the sending unit, select
2nd
VAR-LINK
.
(Link).
2 (Receive).
1 (Send).

This is not a function,
but it would still be
nice to be able to find
the slope.
x2  y 2  1
d 2 d 2 d
x 
y  1
dx
dx
dx
Do the same thing to both sides.
Note use of chain rule.
dy
2x  2 y
0
dx
dy
2y
 2 x
dx
dy 2 x

dx 2 y
dy
x

dx
y

2 y  x 2  sin y
d
d 2 d
2y 
x  sin y
dx
dx
dx
This can’t be solved for y.
dy
2x

dx 2  cos y
dy
dy
2  2 x  cos y
dx
dx
dy
dy
2  cos y
 2x
dx
dx
This technique is called
implicit differentiation.
1 Differentiate both sides w.r.t. x.
dy
 2  cos y   2 x
dx
dy
2 Solve for
.
dx

Find the equations of the lines tangent and normal to the
curve
x 2  xy  y 2  7 at (1, 2) .
We need the slope. Since we can’t solve for y, we use
dy
implicit differentiation to solve for
.
dx
x 2  xy  y 2  7
Note product rule. dy
dy
 dy

2x   x  y  2 y
0
dx
 dx

dy
dy
2x  x  y  2 y
0
dx
dx
dy
 2 y  x  y  2x
dx
y  2x

dx 2 y  x
2  2  1
22
4
m


2  2   1 4  1
5

Find the equations of the lines tangent and normal to the
curve
4
m
5
x 2  xy  y 2  7 at (1, 2) .
tangent:
normal:
4
y  2   x  1
5
5
y  2    x  1
4
4
4
y2 x
5
5
5
5
y2  x
4
4
4
14
y  x
5
5
5
3
y   x
4
4

Higher Order Derivatives
d2y
3
2
Find
if
2
x

3
y
7 .
2
dx
2
y

2
x

x
y
3
2
y 
2x  3 y  7
y2
6 x  6 y y  0
2
2x x 2
y 
 2 y
y y
6 y y  6 x 2
6 x 2
y 
6 y
2
x
y 
y
2x x 2 x 2
y 
 2
y y y
2x x 4
y 
 3
y y
Substitute y
back into the
equation.

Now let’s do one on the TI-89:
2
d
y
3
2
2 x  3 y  7 Find
.
2
dx
APPS
Select
Calculus Tools and press
ENTER
.
If you see this
screen, press
, change the
mode settings as
necessary, and
ENTER
press
again.
APPS

Now let’s do one on the TI-89:
2
d
y
3
2
2 x  3 y  7 Find
.
2
dx
APPS
Press
Select
Calculus Tools and press
Press
F2
4
(Deriv)
(Implicit Difn)
Enter the equation.
(You may have to unlock
the alpha mode.)
Set the order to 2.
Press
ENTER
.
ENTER
.
Press ESC and then
HOME to return your
calculator to normal.
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