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Unformatted text preview: Lecture 16 Lecture 16: (Sec.3.6)
Implicit Diﬀerentiation
dy
for the following:
dx
1
1) y =
2x − 3
Find 2) xy + 2y − x = 0 But consider the following equation:
3) y − xy 2 + 2x = 1 1 Lecture 16 2 Recall the following form of the Chain Rule:
If u is a diﬀerentiable function of y where y is a differentiable function of x, then Now ﬁnd the following for y = f (x):
d3
1) (y )
dy d3
2) (y )
dx d2
3) (x + y 3)
dx 4) d2
(x y )
dx Lecture 16 3 dy
for the equation
dx
2
y − xy + 2x = 1. ex. Find To ﬁnd
1) 2) dy
by Implicit Diﬀerentiation:
dx Lecture 16 4 dy
given that x and y are related by the
ex. Find
dx
equation
x 2 y 3 + 2 − 3y = x 2 . Lecture 16 5 ex. Find the slope of the tangent line to the curve
1
x
= x + 2y at (1, ).
y
2 NOTE: Lecture 16 6 ex. Find the equation of the tangent line to the
curve (x + y )2 − xy 2 = y + 3 at (−1, 2). Lecture 16 7 ex. Find the slope of √ tangent line to the graph
the
of x2 + y 2 = 9 at (2, − 5): 1) Explicitly Lecture 16 8 2) Find the slope of the tangent line to the graph of
√
2
2
x + y = 9 at (2, − 5) using implicit diﬀerentiation. Lecture 16 3) Show that the two forms of d2y
4) Find 2 if x2 + y 2 = 9.
dx 9 dy
are equivalent.
dx Lecture 16 10 ex. Suppose that x and y are diﬀerentiable functions
of t so that x2 + y 2 = 9.
Find dx
dy
in terms of x, y, and .
dt
dt Lecture 16 11 dx
if x2 + y 2 = 9, and y is decreasing at
ex. Find
dt
a constant rate of 0.5 units/second when x = 1 and
y > 0. Lecture 16 12 ex. Suppose a point is moving along the graph of
xy 2 + 2y = −4 so that its xcoordinate is increasing
at the rate of 1.5 inches per minute. How fast is the
y coordinate changing when y = −1? ...
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 Spring '08
 Smith
 Calculus, Chain Rule, The Chain Rule

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