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

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

4. Implicit Differentiation
Trying to find
dy
dx

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

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

7. Example: Find the derivative
x 3 − 2 x 2 y + 3 xy 2 = 38

dy 2   2
dy 
2
3x −  4xy + 2x ÷+  3y + 2y 3x ÷= 0

 
dx
dx 
dy 2
dy
2
2
3x − 4xy − 2x + 3y + 2y 3x = 0
dx
dx
dy
2 dy
−6xy + 2x
= 3x 2 − 4xy + 3y 2
dy dx
2dx
2
2
−6xy + 2x = 3x − 4xy + 3y
dx
(
)
dy 3x − 4xy + 3y
=
dx
−6xy + 2x 2 )
(
2
2

8. Example: Determine the slope at the point (1,1)
x + y = 2 xy
3
3
dy
dy
3x + 3y
= 2y + 2x
dx
dx
dy
2 dy
3y
− 2x
= 2y − 3x 2
dx
dx
2
2
dy
3y 2 − 2x ) = 2y − 3x 2
(
dx
dy
2y − 3x
=
dx ( 3y 2 − 2x )
2
dy
dx
x=1
y=1
dy
dx
x=1
y=1
2 × 3× 2
1− 1
=
3× 2 − 2 × )
( 1 1
=
−1
= −1
1

9. x + y = 2 xy
3
3
dy
2y − 3x
=
2
dx ( 3y − 2x )
2
dy
dx
x=1
y=1
−1
=
= −1
1

10. x + y = 2 xy
3
3
dy
2y − 3x
=
2
dx ( 3y − 2x )
2
dy
dx
x=1
y=1
−1
=
= −1
1