2. Implicit & Explicit Forms
Implicit Form Explicit Form Derivative
Explicit: y in terms of x
Implicit: y and x together
Differentiating: want to be able to use either
1

xy 1
1 

 x
x
y 2
2 1
x
x
dx
dy



 

3. Differentiating with respect to x
Derivative →
Deriving when denominator agrees → use properties
Deriving when denominator disagrees → use chain rule & properties
dx
d
 
2
4x
dx
d
x
8 Denominator agrees -
properties
 
2
3y
dx
d
dx
dy
y
6 Denominator disagrees –
chain rule

4. Implicit Differentiation
Trying to find
dx
dy

5. Derive Explicitly Derive Implicitly
x2
+ y2
= 5 x2
+ y2
= 5
y = ± 5- x2
dy
dx
=
1
±2 5- x2
× -2x
( )
dy
dx
=
-x
± 5- x2
dy
dx
=
-x
y
since y = ± 5- x2
2x+2y
dy
dx
=0
2y
dy
dx
=-2x
y
dy
dx
=-x
dy
dx
=
-x
y

6. Derive implicitly:
y = 3xy4
dy
dx
= 3× y4
+4y3 dy
dx
3x
dy
dx
- 4y3 dy
dx
×3x = 3× y4
dy
dx
1- 4y3
3x
( )= 3× y4
dy
dx
1-12xy3
( )=3×y4
dy
dx
=
3×y4
1-12xy3
( )

7. Example: Find the derivative
38
3
2 2
2
3


 xy
y
x
x
3x2
- 4xy+
dy
dx
2x2
æ
è
ç
ö
ø
÷+ 3y2
+2y
dy
dx
3x
æ
è
ç
ö
ø
÷= 0
3x2
- 4xy-
dy
dx
2x2
+3y2
+2y
dy
dx
3x = 0
-6xy
dy
dx
+2x2 dy
dx
= 3x2
- 4xy+3y2
dy
dx
-6xy+2x2
( )= 3x2
- 4xy+3y2
dy
dx
=
3x2
- 4xy+3y2
-6xy+2x2
( )

8. Example: Determine the slope at the point (1,1)
xy
y
x 2
3
3


3x2
+3y2 dy
dx
= 2y+
dy
dx
2x
3y2 dy
dx
-2x
dy
dx
= 2y -3x2
dy
dx
3y2
-2x
( )= 2y -3x2
dy
dx
x=1
y=1
=
-1
1
= -1
dy
dx
=
2y -3x2
3y2
- 2x
( )
dy
dx
x=1
y=1
=
2×1-3×12
3×12
- 2×1
( )

9. xy
y
x 2
3
3


dy
dx
=
2y -3x2
3y2
- 2x
( )
dy
dx
x=1
y=1
=
-1
1
= -1