This document discusses implicit and explicit differentiation. It provides examples of taking the derivative of equations in both implicit and explicit form. It also shows how to find the derivative at a point, such as finding the slope of an implicitly defined equation at the point (1,1).

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