Analyze
correctly
and
solve
properly
application
problems
concerning
the
derivatives
to
include
writing
equation
of
tangent/normal line, curve tracing ( including all types of algebraic
curves
and
cusps),
optimization
problems,
rate
of
change
and
related-rates problems (time-rate problems).

APPLICATIONS of the DERIVATIVES

LESSON 4:
RELATED RATES and TIME RATES PROBLEMS

OBJECTIVE:
At the end of the lesson, the student would
be able to illustrate and solve related rate
problems including time rate problems.

Recall: The Derivative of a function
When the concept of the derivative was introduced in the earlier discussion, it was
defined as follows:
DEFINITION:
The derivative of
at point P on the curve is equal to the slope of the tangent
line at P; thus the derivative of the function
with respect to
x
at any
x
in its
domain is defined as:
)
(
x
f
y
)
(
x
f
y
0
0
(
)
( )
lim
lim
x
x
dy
y
f x
x
f x
dx
x
x
provided the limit exists.

))
(
,
(
1
1
x
f
x
P
))
(
,
(
2
2
x
f
x
Q
)
(
x
f
y
x
x
x
x
x
x
1
2
1
2
y
secant line
x
y
In the given figure, we note that the line connecting points P and Q is a secant line of the curve
with slope
𝑚 =
Δ?
Δ?
, which gives the average change of
y
per unit change in
x
. However, as Q gets
closer and closer to P ,
∆?
decreases and approaches zero.

))
(
,
(
1
1
x
f
x
P
))
(
,
(
2
2
x
f
x
Q
)
(
x
f
y
x
x
x
x
x
x
1
2
1
2
y
secant line
x
y
The closer Q gets to P, the smaller
∆x
becomes (approaching 0) and the ratio
Δy
Δx
, which gives the
average change of
y
per unit change in
x ,
would define the instantaneous change in
y
per unit
change in
x
; that is
lim
∆?→0
∆?
∆?
= instantaneous rate of change of
y
with respect to
x.

))
(
,
(
1
1
x
f
x
P
))
(
,
(
2
2
x
f
x
Q
)
(
x
f
y
x
x
x
x
x
x
1
2
1
2
y
secant line
x
y
The derivative of a function y = f(x)
is the instantaneous rate of change of
y
with respect to x.

A function may be single-variable,
? = 𝑓(?)
,or multi-variable,
? = 𝑓(?, ?, … )
. Regardless of the
function being single-variable or multi-variable, at times, there is a need to investigate how the function would
change in relation to changes in its independent variable/variables or how fast the dependent variable would
change at the particular instant the independent variable or variables assume a particular value ( instantaneous
rate of change of the dependent variable with respect to its independent variables; general related-rate
problems).
Also, there are many functions in which the concern is on knowing how fast the variables ( both
dependent and independent variables) are changing with respect to time
(time-rate problems). It is not
necessary to express each of these variables directly as function of time. For example, we are given an equation
involving the variables x and y, and that both x and y are functions of the third variable t, where t denotes time.

Both sides of the equation can be differentiated with respect to time
t ,
applying
implicit
differentiation
and the
chain rule

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- Fall '19