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.