CEGEP CHAMPLAIN - ST. LAWRENCE
201-NYA-05: Differential Calculus
Patrice Camiré
Implicit Differentiation
1. Find the derivative of the curve.
(a) x5 + y 5 = 2
(c) x − xy 2 + 2y = y − x
(b) x3 + x2 y + 4y 2 = 6
(d) x2 y + xy 3 = 3x + π
(e) 2y 7 + x3 y 2 = 1 − x2 y
(f)
√
x + y = x2 y 3 − x
2. Find the derivative of the curve and the equation of the tangent line to the curve at the given point.
(a) curve: x2 − 3y 2 = 2x
point: (−1, 1)
(b) curve: 1 + x3 − xy = y 2 + 5x
point: (−2, 3)
(c) curve: x3 + x2 y − y 2 = 1 − 3x + x4
point: (1, 2)
(d) curve: xy − x3 y 2 + 22 = y 3 − x
point: (3, 1)
(e) curve: x2 y 3 + 5x − y = 1 − xy
point: (4, −1)
(f) curve: x2 − x2 y 3 + 1 = 2x5 − y 4
point: (1, 1)
(g) curve: y 8 − 2xy 3 + 1 = x4 + y 5
point: (−1, −1)
(h) curve: 3x2 − xy + y 6 = x + 2y + 3
point: (0, −1)
3. Find the equation of the tangent line to the curve at the given point.
(a) x2 + xy + y 2 = 3 , (1, 1)
(b) x2 + 2xy − y 2 + x = 2 , (1, 2)
(c) 2(x2 + y 2 )2 = 25(x2 − y 2 ) , (3, 1)
(d) y 2 (y 2 − 4) = x2 (x2 − 5) , (0, −2)
√
(e) xy + xy = 2 , (1/2, 2)
(f) |x| + |y| = 1 , (1/2, −1/2)
4. Find all points on the curve for which the tangent line is (i) horizontal and (ii) vertical.
(a) x2 − xy − y 2 = −20
(e) x2 − y 2 − xy + 5x = −10
(b) x2 − xy + y 2 = 75
(f) x2 − y 2 − xy + 5y = 10
(c) x2 + y 2 − xy + 3x = 24
(g) x2 + y 2 − 2x − 2y + 1 = 0
(d) x2 + y 2 − xy + 6y = 36
(h) x2 + y 2 + 2x − 4y + 4 = 0
Answers
dy
x4
=− 4
dx
y
(d)
(b)
dy
x (3 x + 2 y)
=−
dx
x2 + 8 y
(e)
(c)
dy
−2 + y 2
=−
dx
−1 + 2 xy
1. (a)
2. (a)
x−1
dy
=
dx
3y
dy
xy (3 xy + 2)
=−
dx
14 y 6 + 2 x3 y + x2
√
√
dy
−1 + 4 xy 3 x + y − 2 x + y
√
(f)
=−
dx
−1 + 6 x2 y 2 x + y
(e)
2
1
tangent line: y = − x +
3
3
(b)
(c)
(d)
dy
3x2 − y − 5
=
dx
x + 2y
tangent line: y = x + 5
(f)
dy
4x3 − 3x2 − 2xy − 3
=
dx
x2 − 2y
tangent line: y = 2x
(g)
dy
3x2 y 2 − y − 1
=
dx
x − 2x3 y − 3y 2
43
25
tangent line: y = − x +
54
18
3. (a) y = −x + 2
7
3
(b) y = x −
2
2
9
40
(c) y = − x +
13
13
4. (a) (i) (2, 4), (−2, −4)
(ii) ∅
dy
2 xy + y 3 − 3
=−
dx
x (x + 3 y 2 )
(h)
dy
2xy 3 + y + 5
=
dx
1 − x − 3x2 y 2
4
67
tangent line: y = x −
51
51
dy
2x − 2xy 3 − 10x4
=
dx
3x2 y 2 − 4y 3
tangent line: y = 10x − 9
4x3 + 2y 3
dy
= 7
dx
8y − 5y 4 − 6xy 2
6
1
tangent line: y = x −
7
7
1 + y − 6x
dy
= 5
dx
6y − x − 2
tangent line: y = −1
(d) y = −2
(e) y = −4x + 4
(f) y = x − 1
(e) (i) (−1, 3), (−3, −1)
(ii) ∅
(b) (i) (−5, −10), (5, 10)
(ii) (10, 5), (−10, −5)
(f) (i) ∅
(ii) (−1, 3), (3, 1)
(c) (i) (1, 5), (−5, −7)
(ii) (4, 2), (−8, −4)
(g) (i) (1, 0), (1, 2)
(ii) (2, 1), (0, 1)
(d) (i) (2, 4), (−6, −12)
(ii) (6, 0), (−10, −8)
(h) (i) (−1, 1), (−1, 3)
(ii) (0, 2), (−2, 2)