Calculus 221 worksheet Example 1. dy , given Find dx

Implicit differentiation

(3xy + 7)2 = 6y.

Solution: Take the derivative with respect to x of each side of the equation. d d (3xy + 7)2 = 6y dx dx dy dy 2(3xy + 7)3(x + y) = 6 dx dx dy dy (18x2 y + 42x) + (18xy 2 + 42y) = 6 dx dx
dy 2 18x y + 42x − 6 = −(18xy 2 + 42y) dx 18xy 2 + 42y dy =− dx 18x2 y + 42x − 6 3xy 2 + 7y =− 2 . 3x y + 7x − 6 Example 2. dy Find , given dx

y sin(x2 ) = x sin(y 2 ).

Solution: Take the derivative with respect to x of each side of the equation. d d y sin(x2 ) = x sin(y 2 ) dx dx dy dy y cos(x2 )2x + sin(x2 ) = x cos(y 2 )2y + sin(y 2 ) dx dx
dy sin(x2 ) − 2xy cos(y 2 ) = sin(y 2 ) − 2xy cos(x2 ) dx sin(y 2 ) − 2xy cos(x2 ) dy = . dx sin(x2 ) − 2xy cos(y 2 ) Example 3. dy Find , given dx

√

xy = 1 + x2 y.

Solution: 1

2

Take the derivative with respect to x of each side of the equation. d√ d (1 + x2 y) xy = dx dx   dy dy 1 −1/2 x + y = 2xy + x2 (xy) 2 dx dx dy x dy y + √ = 2xy + x2 √ 2 xy dx 2 xy dx   dy y x √ − x2 = 2xy − √ dx 2 xy 2 xy y 2xy − √ 2 xy dy = x dx √ − x2 2 xy =

4(xy)3/2 − y √ . x − 2x2 xy

Exercise 2x − y . x + 3y

1. (3xy + 7)2 = 6y.

2. x3 =

3. xy = cot(xy).

4. x4 + sin y = x3 y 2 .

5. x cos(2x + 3y) = y sin x. 6. Find the tangent line of the curve at the point: x2 −

√

√ 3xy + 2y 2 = 5 at ( 3, 2).

7. Find the tangent line of the curve at the point: x sin 2y = y cos 2x at (π/4, π/2). 8. The line that is normal to the curve x2 + 2xy − 3y 2 = 0 at (1, 1) intersects the curve at what other point? Answer 1.

dy 3xy 2 + 7y = . dx 1 − 3x2 y − 7x

2.

dy 2 − 4x3 − 9x2 y = . dx 3x3 + 1

3.

dy y =− . dx x

4.

dy 3x2 y 2 − 4x3 = . dx cos y − 2x3 y

5.

dy cos(2x + 3y) − y cos x − 2x sin(2x + 3y) = . dx sin x + 3x sin(2x + 3y)

6. y = 2. 8. (3, −1).

7. y = 2x.