MATH 1142
Section 3.3 Worksheet
NAME
Implicit Differentiation
So far we have used what we know about the derivative to calculate the instantaneous rate of change of lots of
functions but we may want to be able to calculate this value for curves that are not functions. For example,
consider
x2 + y 2 = 4.
What does this curve look like?
Is this curve a function?
Notice though that in a small region near almost all the points on the curve that the curve looks like that graph
of a function.
Thus we say that the a function is implicitly defined by the equation. We then get a formula for
dy
dx
by:
There is one thing that you must keep in mind is that when you take the derivative of a function if x
with respect to x you have to use the Chain Rule meaning...
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Use implicit differentiation to compute
dy
dx
for x2 + y 2 = 4.
√
√
Find the slope of the curve at the points (1, 3) and (1, − 3).
Finding
dy
dx
by Implicit Differentiation
1. Differentiate each term of the equation with respect to x treating y as a function of x.
2. Solve for
dy
dx :
• Move all terms involving
• Factor out
dy
dx
dy
dx
to the left and all other terms of the right side of the equation.
on the left had side.
• Divide both sides of the equation by the factor that multiplies
Examples:
1. Use implicit differentiation to calculate
dy
dx
for the following curves.
(a) x4 + (y + 3)4 = x2
2
dy
dx .
(b) x3 y + xy 3 = 4
2. Use implicit differentiation to determine the slope of the graph of y 2 = 3xy − 5 where y = 1.
Related Rates
In many real-life situations when we may have two variables say x and y both treated as functions of some third
variable, say t. When we differentiate the equation with respect to t, we derive a relationship between the rates of
dx
change dy
dt and dt . We say that these are related rates.
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Examples:
1. Suppose that x thousand units of a commodity can be sold weekly when the price is p dollars per unit and
that x and p satisfy the demand equation
p + 2x + xp = 38.
How fast are weekly sales changing at a time when x = 4, p = 6 and the price is falling at a rate of $.40 per
week?
2. A point is moving along the graph of x2 −4y 2 = 9. When the point is at (5, −2), its x-coordinate is increasing
at the rate of 3 units per second. How fast is the y-coordinate changing at that moment?
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