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Lecture 11:
Implicit Differentiation, Related Rates and
Differentials
Review of Lecture 10
Generalized Derivative Formulas
Generalized Derivative Formulas
!
If we let
then we can rewrite the chain rule formula as
This result, called the generalized derivative formula for f
Implicit Differentiation
!
Find dy/dx if
Method 1:
Explicit Differentiation
Method 2:
Implicit Differentiation
Example
Find the slopes of the tangent lines to the curve
at the points
and
(2, 1).
3.8 Related Rates
Liquid draining through a conical filter:
To find
at a specific time t we need values for r, h,
related rates problem
In this case it is clear that from the geometry of the cone
r = h tan α
Example Suppose that x and y are differentiable functions of t and are related by the
equation
. Find
at time
if
and
at time
.
!
Solution. Using the chain rule to differentiate both sides of the equation !
with respect to t yields
!
!
!
!
!
Example In Figure 3.8.5 we have shown a camera mounted at a point 3000 ft from the base of a rocket launching
pad. If the rocket is rising vertically at 880 ft/s when it is 4000 ft above the launching pad, how fast must the camera
elevation angle change at that instant to keep the camera aimed at the rocket?
t = number of seconds elapsed from the time of launch
= camera elevation angle in radians after t seconds
h = height of the rocket in feet after t seconds
Solution
=?
An aircraft is flying horizontally at a constant height of 1500 m above a fixed observation point. At a
certain instant the angle of elevation is 30° and decreasing, and the speed of the aircraft is 600 km/hr.
!
(a) How fast is
!
decreasing at this instant? Express the result in units of degrees/s.
(b) How fast is the distance between the aircraft and the observation point changing at this instant?