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Math 151 – Lynch
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Section 3.6 – Implicit Differentiation
Definition. Instead of defining y = f (x) as an explicit function of x, we can define the
relationship between x and y implicitly by an equation relating x and y.
2
2
Example 1. Consider the circle
√ x + y = 36. Find the slope of the tangent line to
this circle at the point (3, −3 3).
Example 2. Consider the folium of Descartes x3 + y 3 = 6xy. Find the slope of the
tangent line at the point (3, 3)
Implicit Differentiation. If a relationship between two variables x and y is defined
implicitly by an equation, then we can use the method of implicit differentiation to
find the derivative of one variable with respect to the other.
To find
dy
:
dx
1. Differentiate both sides of the equation with respect to x ( remember the derivative
dy
and use the chain rule when necessary).
of y is
dx
2. Solve the resulting equation for
Example 3. Find the
√
the point (3, −3 3).
dy
dx
dy
.
dx
for x2 + y 2 = 36. Then find the tangent line to the circle at
Example 4. Find a formula for
to the equation at (3, 3).
dy
dx
for x3 + y 3 = 6xy. Use it to find the tangent line
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Math 151 – Lynch
Example 5. Find
(a)
3.6–Implicit Differentiation
dy
dx .
1
1
+√ =4
2
x
y
(b) x2 y − 3xy 2 + 5y 3 = 4xy
Example 6. If x
p
3
f (x) − (f (x)) = 6x and f (−4) = 4, then find f 0 (−2).
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Math 151 – Lynch
Example 7. Find
dx
dy
3.6–Implicit Differentiation
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for x4 cos y − x2 = y 3 (sin x − 5y)
Example 8. Find the tangent line to the equation x2 − 3xy − y 2 x = 4 at the point
(2, −3).
Definition. Two curves are called orthogonal if at each point of intersections their
tangent lines are perpendicular.
Example 9. The equation xy = c, c 6= 0 represents a family of hyperbolas. The
equation x2 − y 2 = k, k 6= 0 represents another family of hyperbolas with asymptotes
y = ±x. Show that every curve of the first family is orthogonal to every curve from the
second family. We call two such families of functions orthogonal trajectories.