The Derivative

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Chapter 3
The Derivative
By Kristen Whaley
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3.1Slopes and Rates of Change

• Average Velocity
• Instantaneous Velocity
• Average Rate of Change
• Instantaneous Rate of Change

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Average Velocity

• For an object moving along an s-axis, with s
  f(t), the average velocity of an object between
  times t0 and t1 is

Secant Line the line determined by two points on
a curve
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Instantaneous Velocity

• For an object moving along an s-axis, with s
  f(t), the instantaneous velocity of the object at
  time t0 is

http//www.coolschool.ca/lor/CALC12/unit2/U02L01/a
veragevelocityvsinstantaneous.swf
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Average and Instantaneous Rates of Change

• Slope can be viewed as a rate of change, and can
  be useful beyond simple velocity examples.

• If y f(x), the average rate of change over the
  interval x0, x1 of y with respect to x is

• If y f(x), the instantaneous rate of change of y
  with respect to x at x0 is

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Examples!!

• 1 Find the slope of the graph of f(x) x21 at
  the point x0 2

Were looking for the instantaneous rate of
change (slope) of f(x) at x 2
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Examples!!

• 2 During the first 40s of a rocket flight, the
  rocket is propelled straight up so that in t
  seconds it reaches a height of s5t3 ft.
• How high does the rocket travel?
• What is the average velocity of the rocket during
  the first 40 sec?
• What is the instantaneous velocity of the rocket
  at the 40 sec mark?

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Examples!!

• 2 (cont)
• How high does the rocket travel?

Knowns s 5t3 ft t 40 sec
s 5 (40)3
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Examples!!

• 2 (cont)
• What is the average velocity of the rocket during
  the first 40 sec?

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Examples!!

• 2 (cont)
• What is the instantaneous velocity of the rocket
  at the 40 sec mark?

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3.2The Derivative

• Definition of the derivative
• Tangent Lines
• The Derivative of f
• with Respect to x
• Differentiability
• Derivative Notation
• Derivatives at the
• endpoints of an interval

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Definition of the Derivative

• The derivative of f at x x0 is denoted by
• f (x0) lim f(x1) f(x2)
• x1 x2

x1 ?x2
Assuming this limit exists, f (x0)
the slope of f at (x0, f(x0))
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Tangent Lines

• The tangent line to the graph of f at (x0, f(x0))
  is the line whose equation is

• y - f(x0) f(x0) ( x - x0 )

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The Derivative of f with Respect to x

• f (x) lim f(w) f(x)
• w x

w ? x
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Differentiability

• For a given function, if x0 is not in the domain
  of f, or if the limit does not exist at x0, than
  the function is not differentiable at x0

NOTE If f is differeniable at xx0, then f must
also be continuous at x0

• The most common instances of nondifferentiability
  occur at a

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Derivative Notation
the derivative of f(x) with respect to x
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Derivatives at the Endpoints of an Interval

• If a function f is defined on a closed interval
  a, b, then the derivative f(x) is not defined
  at the endpoints because

f (x) lim f(w) f(x)
w x
w?x
is a two-sided limit. Therefore, define the
derivatives using one-sided, right and left hand,
limits
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Derivatives at the Endpoints of an Interval

• A function f is differentiable on intervals
• a, b
• a, 8)
• (-8, b
• a, b)
• (a, b
• if f is differentiable at all numbers inside the
  interval, and at the endpoints (from the left or
  right)

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Examples!!

• 1 Given that f(3) -1 and f(x) 5, find an
  equation for the tangent line to the graph of y
  f(x) at x3

KNOWNS F(x) slope of the tangent line
5 Point given (3, -1)
USING POINT SLOPE FORM y 1 (5) (x 3)
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Examples!!

• 2 For f(x)3x2 , find f(x), and then find the
  equation of the tangent line to yf(x) at x 3

KNOWNS f(x) slope of tangent line
(6a) point (3, 27)
POINT SLOPE FORM y 27 (18) (x 3)
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3.3Techniques of Differeniation

• Basic Properties
• The Power Rule
• The Product Rule
• The Quotient Rule

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Basic Properties
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The Power Rule
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The Product Rule
The Quotient Rule
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Examples!!

• 1 Find dy/dx of y (x-3) (x4 7)

Let f(x) (x-3) and g(x) (x4 7)
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Examples!!
Let f(x) 4x 1 and g(x) x2 - 5
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3.4Derivatives of Trigonometric Functions

• Derivatives of the Trigonometric Functions (sinx,
  cosx, tanx, secx, cotx, cscx)

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Derivatives of Trigonometric Functions!
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Examples!!
Solve this using the quotient and product rules
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Examples!!
2 Find y (x) of y x3 sin x 5 cos x
Solve this using the product rule
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3.5The Chain Rule

• Derivatives of Compositions
• The Chain Rule
• An Alternate Approach

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Derivatives of Compositions
If you know the derivative of f and g, how can
you use these to find the derivative of the
composition of f g?
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Chain Rule!

• If g is differentiable at x and f is
  differentiable at g(x), then the composition f
  g is differentiable at x
• If y f(g(x)) and u g(x)
• then y f(u)

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An Alternative Approach

• Sometimes it is easier to write the chain rule
  as

g(x) is the inside function
f(x) is the outside function
The derivative of f(g(x)) is the derivative of
the outside function evaluated at the inside
function times the derivative of the inside
function
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An Alternative Approach

• That is

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An Alternative Approach

• Substituting u g(x) you get

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Examples!!
1 Find dy/dx of y (5x 8)13(x3 7x)12
Use the chain rule, and product rule
dy/dx (5x 8)1312(x3 7x)11(3x2 7)
(x3 7x)1213(5x 8)12(5)
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3.6Implicit Differentiation

• Explicit versus Implicit
• Implicit Differentiation

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Explicit Versus Implicit

• A function in the form y f(x) is said to
  define y explicitly as a function of x because
  the variable y appears alone on one side of the
  equation.
• If a function is defined by an equation in which
  y is not alone on one side, we say that the
  function defines y implicitly

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Explicit Versus Implicit

• Implicit
• yx y 1 x
• NOTE The implicit function can sometimes by
  rewritten into an explicit function
• Explicit
• y (x-1) / (x1)

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Explicit Versus Implicit

• A given equation in x and y defines the function
  f implicitly if the graph of y f(x)
  coincides with a portion of the graph of the
  equation

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Explicit Versus Implicit

• So, for example the graph of x2 y2 1 defines
  the functions
• f1(x) v(1-x2)
• f2(x) -v(1-x2)
• implicitly, since the graphs of these functions
  are contained in the circle x2 y2 1

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Explicit Versus Implicit
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Implicit Differentiation

• Usually, it is not necessary to solve an equation
  for y in terms of x in order to differentiate the
  functions defined implicitly by the equation

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Examples!!

• 1 Find dy/dx for sin(x2y2) x

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Examples!!

• 2 Find d2y/dx2 for x3y3 4 0

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3.7Related Rates

• Differentiating Equations to Relate Rates

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Differentiating Equations to Relate Rates

• Strategy for Solving Related Rates

Step 1 Identify the rates of change that are
known and the rate of change that is to be found.
Interpret each rate as a derivative of a
variable with respect to time, and provide a
description of each variable involved.
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Differentiating Equations to Relate Rates

• Strategy for Solving Related Rates

Step 2 Find an equation relating those
quantities whose rates are identified in Step 1.
In a geometric problem, this is aided by drawing
an appropriately labeled figure that illustrates
a relationship involving these quantities.
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Differentiating Equations to Relate Rates

• Strategy for Solving Related Rates

Step 3 Obtain an equation involving the rates
in Step 1 by differentiating both sides of the
equation in Step 2 with respect to the time
variable.
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Differentiating Equations to Relate Rates

• Strategy for Solving Related Rates

Step 4 Evaluate the equation found in Step 3
using the known values for the quantities and
their rates of change at the moment in question.
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Differentiating Equations to Relate Rates

• Strategy for Solving Related Rates

Step 5 Solve for the value of the remaining
rate of change at this moment.
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Example!!

• 1 Sand pouring from a chute forms a conical
  pile whose height is always equal to the
  diameter. If the height increases at a constant
  rate of 5ft/ min, at what rate is sand pouring
  from the chute when the pile is 10 ft high?

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Example!!
1 (cont.)

• STEP1.

t time h height of conical pile at a given
time V amount of sand in conical pile at a
given time
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Example!!
1 (cont.)

• STEP2.

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Example!!
1 (cont.)

• STEP3.

STEP4.
STEP5.
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Example!!

• 2 A 13-ft ladder is leaning against a wall.
  If the top of the ladder slips down the wall at a
  rate of 2 ft/sec, how fast will the foot of the
  ladder be moving away from the wall when the top
  is 5 ft above the ground?

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Example!!
2 (cont.)

• STEP1.

t time h height of the top of the ladder
against the wall D distance of the foot of the
ladder from the base of the wall
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Example!!
2 (cont.)

• STEP2.

D2 h2 132
D2 h2 169
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Example!!
2 (cont.)

• STEP3.

D2 h2 169
(note at h5, D 12)
STEP4.
STEP5.
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3.8Local Linear Approximation Differentials

• Local Linear Approximation
• Differentials

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Local Linear Approximation

• Linear Approximation may be described informally
  in terms of the behavior of the graph of f under
  magnification if f is differentiable at x0, then
  stronger and stronger magnifications at a point,
  P, eventually make the curve segment containing P
  look more and more like a nonvertical line
  segment, that line being the tangent line to the
  graph of f at P.

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Local Linear Approximation

• A function that is differentiable at x0 is said
  to be locally linear at the point P (x0, f(x0))

As you zoom closer to a point P, the function
looks more and more linear
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Local Linear Approximation

• Assume that a function f is differentiable at x0,
  and remember that the equation of the tangent
  line at the point P (x0, f(x0)) is

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Local Linear Approximation

• Since the tangent line closely approximates the
  graph of f for values of x near x0, that means
  that provided x is near x0, then

This is called the local linear approximation of
f at x0
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Local Linear Approximation

• By rewriting the formula

with ?x x - x0, you get
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Local Linear Approximation

• Generally, the accuracy of the local linear
  approximation to f(x) at x0 will deteriorate as c
  gets progressively farther from x0.

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Differentials

• Early in the development of calculus, the symbols
  dy and dx represented infinitely small
  changes in the variables y and x. The
  derivative dy/dx was thought to be a ratio of
  these infinitely small changes. However, the
  precise meaning is logically elusive.
• Our goal is to define the symbols dy and dx so
  that dy/dx can actually be treated as a ratio

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Differentials

• The variable dx is called the differential of x.
  If we are given a function y f(x) tha is
  differentiable at x x0, then we define the
  differential of f at x0 to be the function of dx
  given by the formula

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Differentials

• The symbol dy is simply the dependant variable of
  this function, and is called the differential of
  y. dy is proportional to dx with constant of
  proportionality f (x0). If dx is not 0, you
  can obtain

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Differentials
Because f (x) is equal to the slope of the
tangent line to the graph of f at the point
(x,f(x)), the differentials dy and dx can be
described as the rise and run of this tangent
line.
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EXAMPLES!!

• 1 Find the local linear approximation of x3 at
  x0 1

y 1 3(x-1)
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EXAMPLES!!

• 2 Use an appropriate local linear approximation
  to estimate the value of v (36.03)

Let f(x)v(x) f (x) (1/2)x-(1/2)
f(36.03) f(36) (1/12)(36.03-36)
f(36.03) 6 (1/12)(.03)
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EXAMPLES!!

• 3 Find the differential dy for y xcosx

dy/dx -xsinx cosx
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BIBLIOGRAPHY!!!

• http//mathcs.holycross.edu/spl/old_courses/131_f
  all_2005/tangent_line.gif
• http//www.coolschool.ca/lor/CALC12/unit3/U03L08/e
  xample_07.gif
• http//www.clas.ucsb.edu/staff/Lee/Tangent20and2
  0Derivative.gif
• http//images.search.yahoo.com/search/images/view?
  backhttp3A2F2Fimages.search.yahoo.com2Fsearch
  2Fimages3Fp3Dproduct2Brule2Bcalculus26ei3DU
  TF-826fr3Dyfp-t-50126x3Dwrtw309h272imgurl
  www.karlscalculus.org2Fqrule_still.gifrurlhttp
  3A2F2Fwww.karlscalculus.org2Fcalc4_3.htmlsize
  15.1kBnameqrule_still.gifpproductrulecalcul
  ustypegifno4tt53oid0077160168cbaf58eiUTF
  -8