Limits, Continuity and the Derivative

1

Limits, Limits, Continuity and Continuity and the Derivativethe Derivative

2

The Derivative and the Slope of a GraphThe Derivative and the Slope of a GraphConsider the case of an object moving along a curve Consider the case of an object moving along a curve from left to right.from left to right.

At the point where the At the point where the object is released, it object is released, it continues in a straight continues in a straight line. This line is called the line. This line is called the tangent linetangent line..

The The slope of the tangent slope of the tangent lineline is the direction of the is the direction of the object at that instant.object at that instant.

So we can think of the slope of a curve as So we can think of the slope of a curve as the slope of the slope of the tangent linethe tangent line to the curve at a given point. to the curve at a given point.

3

The Derivative and the Slope of a GraphThe Derivative and the Slope of a GraphUnlike a straight line, a curve changes direction.Unlike a straight line, a curve changes direction.

So the slope of a curve So the slope of a curve changes from one point to changes from one point to another.another.

Thus we speak of Thus we speak of the the slope of the curve at a slope of the curve at a pointpoint..

The slope of the tangent line to the curve at a point is The slope of the tangent line to the curve at a point is called the Derivative of the curve at that point.called the Derivative of the curve at that point.

4

Using Limits to calculate the derivativeUsing Limits to calculate the derivativeRecall that Recall that the derivative the derivative at the indicated point is at the indicated point is the slope of the tangent the slope of the tangent line (indicated here) at line (indicated here) at that pointthat point..We usually need two We usually need two points on the tangent line points on the tangent line to determine its slope. to determine its slope. However, we only have However, we only have one point (indicated).one point (indicated).

How can we determine the slope of the tangent line How can we determine the slope of the tangent line using only one point?using only one point?

)(, afa

5

Using Limits to calculate the derivativeUsing Limits to calculate the derivativeAn approximate value for An approximate value for the slope of the tangent the slope of the tangent line can be obtained from line can be obtained from the slope of a secant line the slope of a secant line that passes through the that passes through the given point.given point. )(, afa

)(, hafha

a

)(af

ha

h

)( haf

The secant line cuts the curve at points (The secant line cuts the curve at points (a, f(a))a, f(a)) and and ((a+h, f(a+h))a+h, f(a+h)). Note that the difference between the . Note that the difference between the x-x-values is values is hh..The slope of the secant line is therefore:The slope of the secant line is therefore:

hafhaf

ahaafhaf

runrisem )()()()(

sec

This is called the This is called the Difference Quotient. Difference Quotient.

6

Using Limits to calculate the derivativeUsing Limits to calculate the derivative

)(, afa

)(, hafha

a

)(af

ha

h

)( haf

Note that slope of the Note that slope of the cyancyan secant line is closer to the secant line is closer to the slope of the slope of the tangenttangent line, than the original line, than the original greengreen secant secant line that we chose. The value of h is smaller for the line that we chose. The value of h is smaller for the cyan secant line than for the original green secant line.cyan secant line than for the original green secant line.

A better approximation A better approximation to the slope of the tangent to the slope of the tangent line will be achieved if we line will be achieved if we choose a point closer to choose a point closer to the given point. That is, if the given point. That is, if we make we make hh smaller. smaller.

7

Using Limits to calculate the derivativeUsing Limits to calculate the derivative

This gives rise to the following limit formula for the This gives rise to the following limit formula for the slope of the tangent line at the point slope of the tangent line at the point (a, f(a))(a, f(a))::

As the second point gets As the second point gets closer to the given point closer to the given point (that is, as h approaches (that is, as h approaches zero), then the slope of zero), then the slope of the secant line the secant line approaches the slope of approaches the slope of the tangent line.the tangent line.

sec00tan lim)()(lim mh

afhafmhh

This is called the Derivative of This is called the Derivative of f(x)f(x) at the point where at the point where x=ax=a, or , or f f ’’ (a)(a). We say “. We say “ff -prime of -prime of aa.”.”

8

Using Limits to calculate the derivativeUsing Limits to calculate the derivativeSo the derivative of So the derivative of f(x)f(x) at the point at the point (a, f(a))(a, f(a)) is: is:

hafhafaf

h

)()(lim)(0

IMPORTANT! Do not forget that:IMPORTANT! Do not forget that:

The derivative of a function at a point is the slope The derivative of a function at a point is the slope of the tangent line to the curve at the given point.of the tangent line to the curve at the given point.

9

Example:Example:Find the equation of the tangent Find the equation of the tangent line to line to f(x)f(x) at the point (2, 1) if at the point (2, 1) if

hfhff

h

)2()2(lim)2(0

11)(

x

xf

Solution:Solution:To find the equation of the tangent line, we need the To find the equation of the tangent line, we need the slope of the line and a point. The point is slope of the line and a point. The point is (2, 1)(2, 1)..The slope of the tangent line is The slope of the tangent line is f f ‘‘(2)(2), since , since x = 2x = 2 at the at the point of tangency. point of tangency. Using the limit definition of the derivative:Using the limit definition of the derivative:

hhhf

11

1)2(1)2( ;; 1

11

121)2(

f

Continued on the next slide…Continued on the next slide…

10

Example (Cont’d):Example (Cont’d):Find the equation of the tangent line to Find the equation of the tangent line to f(x)f(x) at the point (2, 1) if at the point (2, 1) if

hh

hfhff

hh

11

11

lim)2()2(lim)2(00

11)(

x

xf

Solution (Cont’d):Solution (Cont’d):


hhh

hhh

hf

hh

11

)1(1lim111

11lim)2(

00

hhh

hhhf

hh

11

lim11

11lim)2(00

11

Note that both the numerator & denominator equal Note that both the numerator & denominator equal 00 when when h = 0h = 0 is substituted. So we must simplify the is substituted. So we must simplify the fraction and divide numerator & denominator by fraction and divide numerator & denominator by hh (that is, cancel (that is, cancel hh). Then substitute for ). Then substitute for hh and simplify. and simplify.

11

11

Example (Cont’d):Example (Cont’d):Find the equation of the tangent line to Find the equation of the tangent line to f(x)f(x) at the point (2, 1) if at the point (2, 1) if

11)(

x

xf


101

11

1lim11

lim)2(00

hhh

hfhh

11

11

So the slope of tangent line So the slope of tangent line m = – 1m = – 1..Equation of tangent line through point (2, 1) and slope Equation of tangent line through point (2, 1) and slope m = – 1m = – 1 is: is:

21111 xyxxmyy

312 xyxy

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The Derivative FunctionThe Derivative FunctionThe The derivative functionderivative function (or just (or just the derivativethe derivative) is a ) is a function that can provide the slope of the tangent function that can provide the slope of the tangent line for a givenline for a given x- x-value:value:

axatxfaf )()(

Example:Example: The derivative function for The derivative function for

So the derivative at point (3, 9) isSo the derivative at point (3, 9) isxxfisxxf 2)()( 2

6)3(2)3( f

Check if this is so using the limit definition of the Check if this is so using the limit definition of the derivative at derivative at x = x = 3. That is, determine3. That is, determine

hfhff

h

)3()3(lim)3(0

13

The Derivative FunctionThe Derivative FunctionTo calculate the derivative function, simply replace To calculate the derivative function, simply replace aa with with xx in the limit definition of derivative, then in the limit definition of derivative, then solve for a function of solve for a function of xx. That is, calculate:. That is, calculate:

Example:Example: Determine the derivative of Determine the derivative of 2)( xxf


hxfhxfxf

h

)()(lim)(0

h

xhxh

xhxh

xfhxfxf

hh

h

00

0

lim22lim

)()(lim)(

Solution:Solution:

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Example (Cont’d):Example (Cont’d):Find Find forfor


2)( xxf)(xf

hxhxxf

h

0lim)(

To cancel To cancel hh, , rationalize the rationalize the numeratornumerator..

Eliminate the Eliminate the limlim notation when you substitute notation when you substitute h = 0h = 0. . The notation remains until the substitution occurs.The notation remains until the substitution occurs.

xhx

xhxh

xhxxfh

0lim)(

xhxhxhx

h

22

0lim

xhxhxhxxf

h

0

lim)( xhxhh

h

0lim

xhxxf

h

1lim)(0 xx

01

xxf

21)( SoSo

Ans.:Ans.:

15

Example (Cont’d):Example (Cont’d):

So, if So, if thenthen2)( xxf xxf

21)(

Now it is easy to determine the slopes for various Now it is easy to determine the slopes for various x-x-values:values:

21

121)1( f ;;

105

55

521

521)5( f

NoteNote that, even though that, even though f(0) = 2f(0) = 2 (that is, (that is, 00 is in the is in the domain of domain of f f ), ),

DNEf02

1)0( (Does Not Exist)(Does Not Exist)

Why?Why?

16

Continuity & The DerivativeContinuity & The DerivativeSince the derivative is a limit (as Since the derivative is a limit (as hh → → 00)), and since , and since the point of tangency must be on the curve, thenthe point of tangency must be on the curve, then

A function does not have a derivative where it A function does not have a derivative where it is not continuous.is not continuous.

In other words…In other words…

If a function is differentiable at If a function is differentiable at x = cx = c, then the , then the function must be continuous at function must be continuous at x = cx = c..

This means:This means: If a function is differentiable (has a derivative) at a If a function is differentiable (has a derivative) at a point or over an interval, then it is continuous at that point or over an interval, then it is continuous at that point or on that interval.point or on that interval.

17

Continuity & The DerivativeContinuity & The DerivativeImportant Note:Important Note:

Although all differentiable functions are Although all differentiable functions are continuous,continuous,

Not all continuous functions are Not all continuous functions are differentiable!differentiable!

This means:This means: It only works one direction:It only works one direction:

IF Differentiable, THEN ContinuousIF Differentiable, THEN Continuous

Not necessarily vice versa.Not necessarily vice versa.

18

Notation for DifferentiationNotation for DifferentiationOne notation for derivative is already familiar:One notation for derivative is already familiar:

)()( xfdxdyxfy

The derivative of The derivative of f(x)f(x) is denoted is denoted f f ‘‘(x)(x)..

Another notation is called Another notation is called Leibniz notationLeibniz notation::

The derivative of The derivative of yy with respect to with respect to xx is is

Other notations for the derivative (with respect to Other notations for the derivative (with respect to xx) ) of a function of a function y = f(x)y = f(x)::

yyDxfdxd

dxdyxf x )()(

dxdy

SoSo

We will use primarily the first three of notations.We will use primarily the first three of notations.

19

Some Rules of DifferentiationSome Rules of DifferentiationQuestion:Question: What is the tangent line to a straight line? What is the tangent line to a straight line?

mxfbmxxf )()(

Answer:Answer: The line itself. That is, a straight line is its The line itself. That is, a straight line is its own tangent line.own tangent line.

Since the derivative is the slope of the tangent line, Since the derivative is the slope of the tangent line, then then the derivative of a straight line (at any point) is the derivative of a straight line (at any point) is the slope of the linethe slope of the line::

Based on this, since a constant is a horizontal straight Based on this, since a constant is a horizontal straight line, then line, then the derivative of a constant is zerothe derivative of a constant is zero::

0)()( xfcxf

In particular, In particular, mxfmxxf )()(

20

Some Rules of DifferentiationSome Rules of DifferentiationQuestion:Question: What is the tangent line to a straight line? What is the tangent line to a straight line?

mxfbmxxf )()(


Answer:Answer: The line itself. That is, a straight line is its The line itself. That is, a straight line is its own tangent line.own tangent line.

Since the derivative is the slope of the tangent line, Since the derivative is the slope of the tangent line, then then the derivative of a straight line (at any point) is the derivative of a straight line (at any point) is the slope of the linethe slope of the line::

Based on this, since a constant is a horizontal straight Based on this, since a constant is a horizontal straight line, then line, then the derivative of a constant is zerothe derivative of a constant is zero::

0)()( xfcxf

In particular, In particular, mxfmxxf )()(

21

Some Rules of DifferentiationSome Rules of DifferentiationExamples:Examples:

4)(34)( xfxxf

0)(5)( xfxf

21)(430

21)( xfxxf

46)(46)( xfxxf1)()( xfxxf

0)()( xfxf

0)(3)( 21 xfxf

Derivative of a Derivative of a constant is 0.constant is 0.

Continue for more rules…Continue for more rules…

22

Some Rules of DifferentiationSome Rules of DifferentiationSimple Power Rule:Simple Power Rule: 1 nn nxx

dxd

nn is any real is any real number.number.

Example:Example: 4155 55)()( xxxfxxf

Constant Multiple Rule:Constant Multiple Rule: )()( xfcxfcdxd

cc is a constant. is a constant.

Example:Example:3

2131

2316)(6)( 3

1 xxxfxxf

Example:Example: 2112

13

21666 2

1

xxx

dxdx

dxd

Continue for more rules…Continue for more rules…xx

332

1

23

Some Rules of DifferentiationSome Rules of DifferentiationSum and Difference Rules:Sum and Difference Rules:

)()()()( xgxfxgxfdxd

““Derivative of a sum Derivative of a sum is the sum of the is the sum of the derivatives.”derivatives.”

5834)( 23

4

ttttf

Example:Example:

)()()()( xgxfxgxfdxd

““Derivative of a Derivative of a difference is the difference is the difference of the difference of the derivatives.”derivatives.”

5834)( 234

dtdt

dtdt

dtdt

dtdtf

5834 234

ttt

01633

16)( 331

tttf 3

1633

163

1

tt


24

Some Rules of DifferentiationSome Rules of DifferentiationProduct Rule:Product Rule:

)()()()()()( xgxfxgxfxgxfdxd

In abbreviated form (which may be easier to remember):In abbreviated form (which may be easier to remember):

Example:Example: DifferentiateDifferentiate

gfgfgfdxd Read:Read: “Derivative of f times g “Derivative of f times g

equals f-prime g plus f g-prime.”equals f-prime g plus f g-prime.”

5323 23 xxxxyStep 1:Step 1: Identify f(x) and g(x)Identify f(x) and g(x) )(xf )(xg

Step 2:Step 2: Determine fDetermine f‘‘(x) and g(x) and g’’(x)(x)xxxf 3)( 3 33)( 2 xxf

532)( 2 xxxg 34)( xxg


25

Some Rules of DifferentiationSome Rules of DifferentiationProduct Rule Example (Cont’d):Product Rule Example (Cont’d):

34353233 322 xxxxxxdxdy

)(xf )(xg

Step 3:Step 3: Substitute the f(x), g(x), fSubstitute the f(x), g(x), f‘‘(x) and g(x) and g’’(x) into the Product (x) into the Product Rule formula, and simplify:Rule formula, and simplify:

)(xg)(xf

151831210 234 xxxxdxdy


xxxx

xxxxxdxdy

91234

15961596

234

2234

26

Some Rules of DifferentiationSome Rules of DifferentiationQuotient Rule:Quotient Rule:

2)()()()()(

)()(

xgxgxfxgxf

xgxf

dxd

In abbreviated form (which may be easier to remember):In abbreviated form (which may be easier to remember):

Example:Example: DifferentiateDifferentiate

2ggfgf

gf

dxd

Read:Read: “Derivative of f over g “Derivative of f over g equals f-prime g minus f g-prime equals f-prime g minus f g-prime divided by g squared.”divided by g squared.”

123

2

3

xxxy

Step 1:Step 1: Identify f(x) and g(x)Identify f(x) and g(x)

)(xf)(xg

Step 2:Step 2: Determine fDetermine f‘‘(x) and g(x) and g’’(x)(x)23)( 3 xxxf 33)( 2 xxf

1)( 2 xxg xxg 2)( Continued on the next slide…Continued on the next slide…

27

Some Rules of DifferentiationSome Rules of DifferentiationQuotient Rule Example (Cont’d):Quotient Rule Example (Cont’d):

)(xf )(xg

Step 3:Step 3: Substitute the f(x), g(x), fSubstitute the f(x), g(x), f‘‘(x) and g(x) and g’’(x) into the (x) into the Quotient Rule formula, and simplify:Quotient Rule formula, and simplify:

)(xg)(xf


22

322

1

223133

x

xxxxxdxdy

2)(xg

22

24224

1

4623333

x

xxxxxxdxdy

22

24

1

346

x

xxxdxdy

28

Some Rules of DifferentiationSome Rules of DifferentiationChain Rule:Chain Rule:The The Chain RuleChain Rule is the differentiation of composition is the differentiation of composition of functions.of functions.

f)(xg

)()(: xgfxgfNotation


Recall that a composition of functions is a “function of Recall that a composition of functions is a “function of a function.” The output of one function (the “inner” a function.” The output of one function (the “inner” function) is the input of the other function (the “outer” function) is the input of the other function (the “outer” function), as indicated below.function), as indicated below.

xx )(xgf

g

29

Some Rules of DifferentiationSome Rules of DifferentiationChain Rule:Chain Rule:

““The derivative of a composition of functions is equal to the The derivative of a composition of functions is equal to the derivative of the outer function with respect to the inner function derivative of the outer function with respect to the inner function (that is, without changing the inner) multiplied by the derivative (that is, without changing the inner) multiplied by the derivative of the inner function.”of the inner function.”

)(xg

Derivative of outsideDerivative of outside

)()()( xgxgfxgfdxd

32 32 xxyExample:Example: 3)( xxf

2232322 xxx

dxdy

Derivative of insideDerivative of inside

Outer functionOuter function

30

Some Rules of DifferentiationSome Rules of DifferentiationAdditional Chain Rule Notation:Additional Chain Rule Notation:

dxdu

dudy

dxdy

:,1& 234

thenxuuyIf Example:Example:

xudxdu

dudy

dxdy 2

34 3

1

3123

12 13

82134

xxxx

31

Some Rules of DifferentiationSome Rules of DifferentiationSpecial Case of the Chain Rule – Special Case of the Chain Rule – The General Power Rule:The General Power Rule:

)()(,)( 1 xfxfndxdythenxfyIf nn

43

1)(2

xx

xsExample:Example: DifferentiateDifferentiate

Solution:Solution:

21

21 43

43

1

43

1)( 2

22

xx

xxxxxs

324321)( 2

32 xxxxs

322432

321

32

43

121)(

23

xx

xx

xxxs

32

Higher-Order DerivativesHigher-Order DerivativesTechnically, what we have been referring to as the Technically, what we have been referring to as the derivative is actually the derivative is actually the first derivativefirst derivative. That is, it is . That is, it is the function obtained when we differentiate a function the function obtained when we differentiate a function onceonce..If we differentiate again, the result is called the If we differentiate again, the result is called the second second derivativederivative. That is, . That is, the the second derivativesecond derivative is the derivative is the derivative of the first derivativeof the first derivative. . Subsequent derivatives are named similarly. For Subsequent derivatives are named similarly. For example, the seventh derivative of a function is example, the seventh derivative of a function is obtained by taking derivatives seven times (obtained by taking derivatives seven times (the the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the derivative of the functionderivative of the function).).

33

Higher Order DerivativesHigher Order DerivativesNotation:Notation:

First Derivative:First Derivative:

)(xfySuppose

)(xfdxdy

Second Derivative:Second Derivative: )(2

2

xfdxyd

Third Derivative:Third Derivative: )(3

3

xfdxyd

Fourth Derivative:Fourth Derivative: )(44

4

xfdxyd

)(xf

dxyd nn

n

:Derivativenth

34

Higher Order DerivativesHigher Order DerivativesExample:Example:

32)(21

ttfiffFind

Solution:Solution: 2

13232)( tttf

21

21

3223221)( tttf

23

23

3223221)( tttf

25

25

32323223)(

tttf

323433

2123

21

252

5

f

35

Implicit DifferentiationImplicit DifferentiationRecall the Chain Rule:Recall the Chain Rule:

Recall also that the Chain Rule applies to a composition Recall also that the Chain Rule applies to a composition of functions: f(g(x)).of functions: f(g(x)).

)()()( xgxgfxgfdxd

73 25 xxySupposeNow, examine the following example:Now, examine the following example:

3253 73 xxyThenxx

dxdy 65 4

?733253 xxyofderivativetheisWhat

3253 73 xxdxdy

dxd

Ans.:Ans.: ……using the Chain Rule…using the Chain Rule…


36

Implicit DifferentiationImplicit DifferentiationExample continued:Example continued:

3253 73 xxdxdy

dxd ……using the Chain Rule:using the Chain Rule:

73733 25225 xxdxdxx

xxxx 65733 4225

dxdyyy

dxd

23 3dxdyy

dxdyy 23

So…So…

Note:Note: The derivative The derivative resemblesresembles a regular derivative a regular derivative An additional dy/dx is multiplied in the derivative. An additional dy/dx is multiplied in the derivative. Why?Why? We use the Chain Rule to do the derivative.We use the Chain Rule to do the derivative.

37

Implicit DifferentiationImplicit DifferentiationA Loose Description of Implicit Differentiation:A Loose Description of Implicit Differentiation:

dxdyyfyf

dxd

)()(

Note:Note: When differentiating keep the following in mind:When differentiating keep the following in mind: Always differentiate BOTH SIDES of the equation with respect Always differentiate BOTH SIDES of the equation with respect to the same variable.to the same variable. The variable that we differentiate with respect to occurs in the The variable that we differentiate with respect to occurs in the denominator of the derivative expression. denominator of the derivative expression. For exampleFor example, if we are , if we are seeking seeking dy/dxdy/dx, then differentiate with respect to , then differentiate with respect to xx. If we are . If we are seeking seeking dV/dtdV/dt, then differentiate with respect to , then differentiate with respect to tt. .

To differentiate a function of y with respect to x: To differentiate a function of y with respect to x:

Differentiate the function as usual (in terms of y), thenDifferentiate the function as usual (in terms of y), then

Multiply by dy/dx .Multiply by dy/dx .

38

Implicit DifferentiationImplicit DifferentiationExample:Example:

332 yxfordxdyFind Differentiate both sides Differentiate both sides

with respect to x. Use the with respect to x. Use the sum/difference rule where sum/difference rule where necessary.necessary.

Steps:Steps:

332

dxdyx

dxd

332

dxdy

dxdx

dxd

Determine whether the Determine whether the term we differentiate term we differentiate contains x or y. If it is a contains x or y. If it is a function of x, then regular function of x, then regular derivatives (since we derivatives (since we differentiate with respect differentiate with respect to x). If it is a function of to x). If it is a function of a variable other than x, (y a variable other than x, (y in this case), then it is in this case), then it is implicit differentiation.implicit differentiation.

This term is a This term is a function of x, function of x, so regular so regular differentiation.differentiation.

This term is a This term is a function of y, function of y, so so ImplicitImplicit differentiation.differentiation.

Note that we differentiate both Note that we differentiate both sides with respect to x.sides with respect to x.

39

Implicit DifferentiationImplicit DifferentiationExample (Cont’d):Example (Cont’d):

Differentiate each term Differentiate each term using the appropriate rules using the appropriate rules of differentiation. of differentiation. Remember, for implicit Remember, for implicit differentiation, differentiate differentiation, differentiate as usual, but multiply by as usual, but multiply by dy/dx at the end.dy/dx at the end.

Steps:Steps:

332

dxdy

dxdx

dxd

Solve the equation for Solve the equation for dy/dx.dy/dx.

032 2 dxdyyx

Regular Regular differen-differen-tiationtiation

Implicit Implicit differen-differen-tiationtiation

Derivative of a Derivative of a constant is 0constant is 0

x2 x2

xdxdyy 23 2

23y23y

232yx

dxdy

40

Implicit DifferentiationImplicit DifferentiationAnother Example:Another Example:

).1,2(

622

at

xyyxfordxdyFind Differentiate both sides Differentiate both sides

with respect to x. Use the with respect to x. Use the sum/difference rule where sum/difference rule where necessary.necessary.

Steps:Steps:

To differentiate a To differentiate a product, use the Product product, use the Product Rule. Be sure to put all x in Rule. Be sure to put all x in one fxn and all y in the one fxn and all y in the other.other.

622 dxdxy

dxdyx

dxd

yxgxxflet

yxdxdFor

)(&)(

,

2

2

)()()()(2 xgxfxgxfyxdxdSo

Continued on next slide…Continued on next slide…

One fxn in terms of xOne fxn in terms of x

One fxn in terms of yOne fxn in terms of y

41

Implicit DifferentiationImplicit DifferentiationExample 1 (Cont’d):Example 1 (Cont’d):

Note that both parts of Note that both parts of the product are in fxns of x: the product are in fxns of x: f(x) & g(x).f(x) & g(x).

Steps:Steps:

Regular differentiation, since f(x) is a Regular differentiation, since f(x) is a fxn of x and we differentiate dx.fxn of x and we differentiate dx.

622 dxdxy

dxdyx

dxd

yxgxxflet

yxdxdFor

)(&)(

,

2

2


xxfxxf 2)()( 2

dxdy

dxdyxgyxg 1)()(

Implicit differentiation, since f(x) is a Implicit differentiation, since f(x) is a fxn of y and we differentiate dx.fxn of y and we differentiate dx.

When doing each When doing each differentiation, be sure to differentiation, be sure to identify whether you need to identify whether you need to do implicit differentiation or do implicit differentiation or regular differentiation.regular differentiation.

42


Complete the Product Complete the Product Rule. Rule. Be careful to Be careful to substitute carefully.substitute carefully.

Steps:Steps:

)()()()(2 xgxfxgxfyxdxd


xxfxxf 2)()( 2

dxdy

dxdyxgyxg 1)()(

Do the same for all Do the same for all products.products.

dxdyxyx 22

dxdyxxy 22

xxgyxflet

xydxdforSimilarly

)(&)(

,,

2

2

dxdyyxf 2)( 1)( xg

Implicit differentiation, Implicit differentiation, since f(x) is a fxn of y and since f(x) is a fxn of y and we differentiate dx.we differentiate dx.

Regular differentiation, Regular differentiation, since g(x) is a fxn of x since g(x) is a fxn of x and we differentiate dx.and we differentiate dx. 12 22 yx

dxdyyxy

dxd 22 y

dxdyxy

43


Substitute all Substitute all derivatives into the derivatives into the original equation.original equation.

Steps:Steps:

Since we wish to find Since we wish to find dy/dx at point (2, –1), dy/dx at point (2, –1), substitute x = 2 & y = –1, substitute x = 2 & y = –1, then solve for dy/dx.then solve for dy/dx.

022 22

y

dxdyxy

dxdyxxy

622 dxdxy

dxdyx

dxd

011222122 22

dxdy

dxdy

01444 dxdy

dxdy

58 dxdy

85

dxdy

AnswerAnswer

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For more examples on how to find Derivatives, For more examples on how to find Derivatives, including Implicit Differentiation, check the including Implicit Differentiation, check the

following websites:following websites:http://www.sosmath.com/calculus/diff/der05/der05.html

andand

http://archives.math.utk.edu/visual.calculus/2/index.html

http://www.sosmath.com/calculus/diff/der05/der05.html

http://archives.math.utk.edu/visual.calculus/2/index.html

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Limits, Continuity and the Derivative

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