Implicit differentiation, Part II

Differentiate x2+4xy+4y2-3x=6

038442 dx

dyy

dx

dyxyx

dx

dyyxyx )84(342

dx

dy

yx

yx

84

342

How do we determine where the slope of the curve is undefined?

Ex. 1 Determine where the slope of x2+4xy+4y2-3x=6 is undefined.

-4x- 8y=0

-4x = 8y

- ½ x = y or x = - 2y

(any point where y = - ½ x)

Steps

1. Find dy/dx

2. Set denominator = 0

3. Solve for x or ydx

dy

yx

yx

84

342

How do we use implicit differentiation to find

Determine of y2 +2y = 2x + 1 Steps

1. Find dy/dx

2. Use implicit differentiation on dy/dx to find

3. Where dy/dx occurs, substitute the dy/dx found in step 1

4. Simplify

2

2

dx

yd

2

2

dx

yd

2

2

dx

yd

222 dx

dy

dx

dyy

2)22( dx

dyy

1

1

ydx

dy

22

2

1

1)0)(1(

y

dxdyy

dx

yd

22

2

)1(

11

10

y

y

dx

yd

32

2

)1(

1

ydx

yd

Find the points on the curve x2 + xy + y2 = 7a) where the tangent is parallel to the y-axis and (undefined)

b) where the tangent is parallel to the x-axis (zero)

022 dx

dyyy

dx

dyxx Steps

1. Find dy/dx

2. If looking for undefined: set denominator = 0

3. If looking for zero slope, set numerator = 0

dx

dyyxyx 22

)2(1

2

yx

yx

dx

dy

0)2(1) yxa

22

xyoryx

02) yxb

xyory

x 22

The line that is normal to the curve x2 + 2xy – 3y2 = 0 at (1,1) intersects the curve at what other point?

Steps

1. Find dy/dx

2. Find the equation of the normal

3. Solve the normal for y

4. Sub y back into the original equation and solve for x

5. Get y to go with x from normal