Download Implicit differentiation Sometimes we are given an equation that

Implicit differentiation
Sometimes we are given an equation that implicitly defines a
relationship between x and y. For example, we could have:
x2 + y 2 = 1.
dy
We’d still like to find dx
.
Differentiate both sides of the equation with respect to x, treating y as if it were a function of x. [Use the Chain Rule, since y
is still a function of x.]
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For example, above we had
x2 + y 2 = 1.
Differentiating both sides with respect to x, we get:
2x + 2y
dy
= 0,
dx
which becomes
dy
−2x
x
=
= .
dx
2y
y
2
Examples:
dy
in the following situations:
Find dx
3x − 5y + 1 = 0
x2 + xy − 5y = 0
x2y − 3xy = x
y 3 − 3x2 + 5x − 1 = xy
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Example
Take an ellipse given by
3x2 + 4y 2 = 16.
Find the equation of the tangent line to this curve at the point
(2, 1).
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Example Suppose that a producer as two sources of raw iron,
x and y, and that for given values (in tons) the amount of steel
that can be produced is
x2 + 3xy + y 2
.
S=
10
Currently, the producer gets 10 tons of iron x and 20 tons of
iron y. Estimate how much more of iron y will be needed if only
9 tons of iron x are available, in order to keep production of steel
constant.
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Related rates
In a related rates problem, we typically have three variables. We
have time t, and then two functions x and y which both depend
on t. We have an equation that x and y satisfy, and we want to
dy
relate dx
and
dt
dt . We use implicit differentiation again.
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Example
Suppose that the area of a circle is growing at a rate of 1.2
cm/s. How fast is the radius growing when the radius is 5cm?
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Example
A ladder 10 feet long is resting against a wall. If the bottom of
the ladder slips, and slides away from the wall at a rate of 1 foot
per second, how fast is the top of the ladder moving down the
wall when the bottom of the ladder is 8 feet from the wall?
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