Implicit Differentiation Worksheet 1

Worksheet Implicit Differentiation Name___________________________________

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use implicit differentiation to find dy/dx. 1) 2xy - y2 = 1 x A) x-y

y B) y-x

y C) x-y

x D) y-x

2) x3 + 3x2y + y3 = 8 x2 + 3xy A) x 2 + y2

x2 + 2xy B) x2 + y2

x2 + 2xy C) x 2 + y2

x2 + 3xy D) x2 + y2

1

1)

2)

3) y x + 1 = 4 2y A) x+1

4) x5 = cot y A) -

5x4

csc y cot y

5) cos xy + x 7 = y7 7x6 + y sin xy A) 7y6 - x sin xy

y B) 2(x + 1)

y C) 2(x + 1)

csc2 y B) 5x4

B)

C) -

7x6 - y sin xy

C)

7y6 + x sin xy

2

5x4

csc2 y

7x6 - x sin xy 7y6

3)

2y D) x+1

D)

D)

4)

5x4

csc2 y

7x6 + x sin xy 7y6

5)

6) x = sec(3y) 1 A) cos(3y) cot(3y) 3

1 B) sec(3y) tan(3y) 3

C) cos(3y) cot(3y)

6)

D) 3 sec(3y) tan(3y)

Use implicit differentiation to find dy/dx and d2 y/dx2 .

7) 2y - x + xy = 3 dy 1 - y d2y 2y - 2 = ; = A) dx 2 + x dx2 (2 + x)2 C)

dy y + 1 d2 y 2y + 2 = ; = dx x + 2 dx2 (x + 2)2

3

d2 y

B)

dy 1+y y+ 1 =; = dx x + 2 dx2 (2 + x)2

D)

dy 1 + y d2 y 2y - 2 =; = dx x + 2 dx2 (x + 2)2

7)

8) x2 - y3 = 9 dy 2x d2 y 6y3 - 8x 2 = ; = A) dx 3y2 dx2 9y6 C)

d2 y

dy 2x 5y3 - 8x2 = ; = B) dx 3y2 dx2 9y5

dy 2x d2 y 6y3 - 8x2 = ; = dx 3y2 dx2 9y3

D)

8)

dy 2x d2 y 6y3 - 8x2 = ; = dx 3y2 dx2 9y5

At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested.

9) y5 + x3 = y2 + 9x, slope at (0, 1) A) - 2

10) x3 y3 = 8, tangent at (2, 1) 1 A) y = - x + 2 2

9 B) 7

C) 3

B) y = 4x - 1

1 C) y = x 2

4

9 D) 5

9)

10) D) y = 4x + 1