This document discusses two related mathematical situations: implicit differentiation and related rates. For implicit differentiation (Situation 1), curves that are not graphs of functions are examined, and the method of implicit differentiation is used to find equations of tangent lines. For related rates (Situation 2), the rate of change of one quantity is determined given the rate of change of a related quantity as described by a formula. Examples are provided for implicit differentiation involving curves like circles, cardioids, and lemniscates. The theory of implicit differentiation is explained, noting that it involves taking the derivative of the equation with respect to x and applying the chain rule since y is implicitly a function of x.

1. IMPLICIT DIFFERENTIATION
AND
RELATED RATES
Once again we are going to look at two separate
and apparently distinct situations that, not
surprisingly, are resolved with the same
mathematical model.
Situation no. 1: Some very nice looking and
useful curves in the plane are NOT describable as
(i.e., graphs of functions)
but rather as
i.e. solution sets of equations.
Compute equations of tangent lines.

2. Situation no. 2: Two quantities and are
related to each other via some formula
(easiest example:
is the surface area of a sphere,
is the volume of the same sphere.
You should be able to show that
, the formula that relates and )
You know how fast changes.
How fast does change?
(Pump air in the sphere at a certain known rate.
How fast does the surface area expand?)
A
V
A V

3. We are going to look at Situation no. 1 first, and
first I will show you some curves that are not
graphs of functions:
the point is
the equation of the
tangent line is ??

4. One more:
the point is
the equation of the
tangent line is ??
This curve is called the
cardioid
One last example, then some theory.

5. Last example:
the point is
(check that it’s on !)
the equation of the
tangent line is ??
This curve is called the
lemniscate
Now to the theory.
Actually it is rather simple. Look at

6. and
1. take a derivative with respect to .
Careful ! You must keep in mind that implicitly,
(by implication from ) ……
is a function of , so the chain rule applies.
That means that you will get an expression with
and and appearing.
Solve for and you got your slope. Since a
point on the curve is given, the point-slope
formula gives you the answer!
Let’s do a couple of example, in fact the 3 we’ve
seen, circle, cardioid, lemniscate.
y x
x

7. I will write the equation on the board and do the
work there.
 Circle
Differentiate to get

8. Cardioid
the answer !!

9. Lemniscate