Differential calculus

April 19, 2023Kalkulus I 1


Coordinate geometryCoordinate geometry

Slope of a curveSlope of a curve

Rapid differentiationRapid differentiation

Higher derivativesHigher derivatives

NotationsNotations

Derivatives ofDerivatives of sum and products;sum and products; of a ’function of a function’ ;of a ’function of a function’ ;

a ratio.a ratio.

Maxima and minimaMaxima and minima Points of inflectionPoints of inflection Sketching curvesSketching curves Partial differentiationPartial differentiation


Geometry

Trigonometry

Coordinate geometry

Relationships between lengths and angles

Use of geometrical insights and

understanding for studying algebraic

problems

-2 -1

210

30

10

20

-20

y

x-10

-2 -1

210

30

10

20

-20

y

x-10

xy 107 xy 107

Algebraic equationAlgebraic equation

Geometrical modelGeometrical model

ordinate ordinate yy

abscissa abscissa xx

origin (0,0)origin (0,0)

The point isThe point is 2 units 2 units RIGHT from y-axis RIGHT from y-axis

and and 27 units UP from 27 units UP from x-axisx-axis

The point isThe point is 2 units 2 units LEFT from y-axis LEFT from y-axis and and 14 units DOWN from 14 units DOWN from

x-axisx-axis


y2

y1

x2

x1

0

y

x

y2

y1

x2

x1

0

y

x

bxay bxay Algebraic equationAlgebraic equation

Geometrical modelGeometrical model

12

12

xx

yyb

12

12

xx

yyb

SlopeSlope

yy22 - - yy11

xx22 - - xx11

Intercept on Intercept on yy-axis-axisy = ay = a

Intercept on Intercept on xx-axis-axis

x = x = ––b b / / aa

The quantity The quantity bb has a meaning has a meaning very similar to very similar to that of the that of the gradient of a hillgradient of a hill in everyday life; in everyday life; the steeperthe steeper the the hill the greater hill the greater gradient, and gradient, and the the more rapidly more rapidly the the height increases height increases with the with the horizontal horizontal distance distance travelled. travelled.


Notation for a change Notation for a change xx22--xx11 in a variable in a variable x:x:

SymbolSymbol MeansMeans

x x Change of any Change of any magnitudemagnitude

xx Small changeSmall changeddxx infinitesimal infinitesimal

changechange (approaching (approaching

zero)zero)xx infinitesimal infinitesimal

changechange under specified under specified

conditioncondition

0 1 2 3 40

10

20

30

y

x0 1 2 3 40

10

20

30

y

x

232 xxy 232 xxy

Algebraic equationAlgebraic equation Geometrical modelGeometrical model

A A positivpositiv

e e change change

of xof x

IncreaIncrease se in in

xxAny Any

change change of x, but of x, but

zerozero

IncremIncrementent inin

xx

If If xx is an increment in is an increment in xx, , then then yy is the is the corresponding corresponding increment in increment in yy, i.e. change in , i.e. change in yy that that occurs as a occurs as a result of result of changechange in in xx


0 1 2 3 40

10

20

30

y

x0 1 2 3 40

10

20

30

y

x

232 xxy 232 xxy


2)(3)(

232

2

xxxxyy

xxy

2)(3)(

232

2

xxxxyy

xxy

xxxxxxyy

xxy

3223

2322

2

xxxxxxyy

xxy

3223

2322

2

xxxxxxxy 2323 2 xxxxxxxy 2323 2xx

x

y

23 xxx

y

23


0 1 2 3 40

10

20

30

y

x0 1 2 3 40

10

20

30

y

x

232 xxy 232 xxy


xxx

y

23 xxx

y

23 xxx

yδ23

δ

δ xx

x

yδ23

δ

δ

Values of the Values of the increment ratioincrement ratio

for for x x = 3= 3

xx yy

11 1010 10100.50.5 4.754.75 9.59.50.20.2 1.841.84 9.29.20.10.1 0.910.91 9.19.10.010.01 0.09010.0901 9.019.010.0010.001 0.0090010.009001 9.0019.0010.00010.0001 0.000900010.00090001 9.00019.00010.000010.00001 0.00009000010.0000900001

9.000019.00001

xδ

δyxδ

δy xx

yx

23δ

δlim

0δ

x

x

yx

23δ

δlim

0δ

xx

y

x

yx

23δ

δlim

d

d0δ

xx

y

x

yx

23δ

δlim

d

d0δ

tangent at tangent at xx = 3 = 3

has has slopeslope 9. 9.

The derivative of The derivative of the function the function xx22+3+3xx+2+2 with with respect to respect to xx is is 3 + 3 + 22xx

Derivative of Derivative of yywith respect to with respect to xx


Some useful relationsSome useful relations

1ii xiAx

y , xAy

d

d function For

1ii xiAx

y , xAy

d

d function For

xx Ax

y , Ay e

d

de function For

xx Ax

y , Ay e

d

de function For

x

A

x

y , xAy

d

d) ln( function For

x

A

x

y , xAy

d

d) ln( function For

0d

d

x

A 0d

d

x

A

A

x

Ax

d

d A

x

Ax

d

d

2d

d

x

A

x

A

x

2d

d

x

A

x

A

x

xx

xee

d

d xx

xee

d

d

x

xx

1 ln

d

d x

xx

1 ln

d

d


232 xxy 232 xxy

032d

d x

x

y 032d

d x

x

y

2d

d3

d

d

d

d

d

d 2

xx

xx

xx

y 2

d

d3

d

d

d

d

d

d 2

xx

xx

xx

y

vuy vuy

vδδδ uy vδδδ uy

xx

u

x

y

δ

δ

δ

δ

δ

δ vxx

u

x

y

δ

δ

δ

δ

δ

δ v

xx

u

x

u

d

d

d

d

d

d vv

xx

u

x

u

d

d

d

d

d

d vv

xx

u

x

y

x

yxxx δ

δlim

δ

δlim

δ

δlim

d

d0δ0δ0δ

v

xx

u

x

y

x

yxxx δ

δlim

δ

δlim

δ

δlim

d

d0δ0δ0δ

v

xx

u

x

u

d

d

d

d

d

d vv

xx

u

x

u

d

d

d

d

d

d vv


vvv δδδδδ uuuy vvv δδδδδ uuuy

deacreases deacreases slower slower

vuy vuy

vv δδδ uuyy vv δδδ uuyy

vvvv δδδδδ uuuuyy vvvv δδδδδ uuuuyy

x

u

xu

x

y

x

yx d

d

dδ

δlim

d

d d0δ

vv

x

u

xu

x

y

x

yx d

d

dδ

δlim

d

d d0δ

vv

x

u

xu

x

y

δ

δ

δ

δ

δ

δ vv

x

u

xu

x

y

δ

δ

δ

δ

δ

δ vv

deacreases deacreases very fastvery fast

uu vv

uu vvuu vv

uu vvuu vv

uu vvuu vv

uu vv

uu vvvv uuy δδδ vv uuy δδδ

uuy DDD vv uuy DDD vv


1

11

Cx

x

BA

CxxBA

y

1

11

Cx

x

BA

CxxBA

y

Cxx

Bu 1 Cx

x

Bu 1

xux

y

x

yx d

d

d

d

δ

δlim

d

d0δ

uy

xux

y

x

yx d

d

d

d

δ

δlim

d

d0δ

uy

CBxx

u 2

d

d CBxx

u 2

d

d

Michaelis functionMichaelis function

xfffx

yxfxfxfy n d

d

d

d

d

d

d

d

d

d)()()( n

3

2

2

1

121

fffy

xfffx

yxfxfxfy n d

d

d

d

d

d

d

d

d

d)()()( n

3

2

2

1

121

fffy

Chain ruleChain ruleChain ruleChain rule


1-uy v1-uy v

x

u

xu

x

y --

d

d

d

d

d

d 12 vv

v x

u

xu

x

y --

d

d

d

d

d

d 12 vv

v

Function of Function of a functiona functionFunction of Function of a functiona function

2dd

dd

d

d

v

vv

xu

xu

x

y 2

dd

dd

d

d

v

vv

xu

xu

x

y

2

DDD

vvv uu

y

2

DDD

vvv uu

y

2d

d

vu-v

x

y2d

d

vu-v

x

y

x

xyxxu

1;1; v

x

xyxxu

1;1; v

1d

d;1

d

d

xx

u v 1d

d;1

d

d

xx

u v

22 1

1

1

1

d

d

xx

xx

x

y

22 1

1

1

1

d

d

xx

xx

x

y


0d

d

d

d...

d

d

d

d

times

x

y

xxxn

0

d

d

d

d...

d

d

d

d

times

x

y

xxxn

Rate of change Rate of change of slopeof slope

(curvature)(curvature)

Rate of change Rate of change of slopeof slope

(curvature)(curvature)First derivativ

eSecond

derivative

Third derivativ

e

Derivative of n-th order

x

y

xxxx

y

n

n

n

d

d

d

d...

d

d

d

d

d

d

times

x

y

xxxx

y

n

n

n

d

d

d

d...

d

d

d

d

d

d

times


Leibnitz notations:Leibnitz notations:

Function: Function: y y

1st derivative:1st derivative:or or DDyy

2nd derivative:2nd derivative:or or DD22yy

n-th derivative:n-th derivative:or or DDnnyy

n

n

x

y

d

dn

n

x

y

d

d

2

2

d

d

x

y2

2

d

d

x

y

x

y

d

dx

y

d

d

Compact notations:Compact notations:Function: Function: y y oror ff((xx))

1st derivative:1st derivative: y'y' or or f f ''((xx))

2nd derivative:2nd derivative: y''y'' or or f f ''''((xx))

n-th derivative:n-th derivative: yy((n)n)or or f f ((n)n)((xx))

Concerns derivative Concerns derivative with respect to x.with respect to x.

Newton’s Newton’s notations:notations:

Function: Function: xx((tt))

1st derivative:1st derivative:

2nd 2nd derivative:derivative:

Concerns Concerns derivatives of derivatives of

time-dependent time-dependent quantities.quantities.

xx

xx


Slope = 0

Maximum of a

function

Not always trueNot always trueNot always trueNot always true

Always trueAlways true

A derivative A derivative shows the slope !shows the slope ! Where a maximimWhere a maximim

occurs ???occurs ???

0 1 2 3 40

10

20

30

v

a0 1 2 3 40

10

20

30

v

a

Substrate inhibition in Substrate inhibition in an enzyme-catalyzed an enzyme-catalyzed reactionreaction

Slope = 0Slope = 0

at the maximumat the maximum


0 1 2 3 40

10

20

30

v

a0 1 2 3 40

10

20

30

v

a

Substrate inhibition in an Substrate inhibition in an enzyme-catalyzed reactionenzyme-catalyzed reaction

Slope = 0Slope = 0

at the maximumat the maximum


If I want to plot If I want to plot yy against log against log xx, , would a maximum appear at the would a maximum appear at the

xx value as in normal plot value as in normal plot ??? ???

Chain rule

The maximum The maximum will be at the will be at the same place !!same place !!


y

x

y

x

'+ 0 + 0 ––' is the sequence is the sequence of signs of of signs of first first derivative around a derivative around a maximummaximum..

'– – 0 +0 +' is the sequence is the sequence of signs of of signs of first first derivative around a derivative around a minimumminimum..

Slope = 0

Maximum of a

function

Not always trueNot always trueNot always trueNot always true

Always trueAlways true

MaximumMaximum

slope = 0slope = 0

MinimumMinimum

slope = 0slope = 0

Function decreasesFunction decreaseswhen x increases,when x increases,

slope < 0slope < 0

Function increasesFunction increaseswhen x increases,when x increases,

slope > 0slope > 0

Function decreasesFunction decreaseswhen x increases,when x increases,

slope < 0slope < 0

A stationary A stationary

pointpoint

A stationary point A stationary point embraces both embraces both maxima and maxima and minima.minima.


0 1 2 3 40

5

10

15y

x0 1 2 3 40

5

10

15y

x

0 1 2 3 4

-5

0

5

10

y'

x0 1 2 3 4

-5

0

5

10

y'

x

0 1 2 3 4

-10

-5

0

5

10

y''

x0 1 2 3 4

-10

-5

0

5

10

y''

x

A A maximum maximum corresponcorresponds to a ds to a zerozero in in first first derivative derivative and and negativenegative second second derivativederivative..

A A minimumminimum corresponcorresponds to a ds to a zerozero in in first first derivative derivative and and positivepositive second second derivativederivative..

A A inflection inflection pointpoint correspondcorresponds to a s to a stacionary stacionary pointpoint in in first first derivative derivative and and zerozero in in second second derivative.derivative.


Points of Points of inflection in inflection in biochemistry biochemistry define define conditions in conditions in which a which a responseresponse (e.g., rate of (e.g., rate of reaction) is reaction) is mostmost (or (or least) least) sensitivesensitive to to an an influenceinfluence (e.g. the (e.g. the concentration concentration of a of a metabolite).metabolite).

Concent.Concent. of a weak acidof a weak acid

Concent.Concent. of a weak acidof a weak acid

xx mol L mol L-1-1 of NaOHof NaOH AA mol Lmol L-1-1

acetic acid, HOAcacetic acid, HOAc

BufferBuffer

Henderson-Hasselbalch equationHenderson-Hasselbalch equation Concent.Concent. of a salt of theof a salt of the

acidacid

Concent.Concent. of a salt of theof a salt of the

acidacid

Negative logarithm ofNegative logarithm ofthe acid dissiciacion the acid dissiciacion

constantconstant

Negative logarithm ofNegative logarithm ofthe acid dissiciacion the acid dissiciacion

constantconstant

The first derivative of The first derivative of pH pH with respect to with respect to x x is a measure of is a measure of the sensitivity of the pH to addition of base. If it is small it the sensitivity of the pH to addition of base. If it is small it means that means that pH pH of the buffer will be not changed much with of the buffer will be not changed much with adding a trace of alkali (an effective buffer). So, the buffer is adding a trace of alkali (an effective buffer). So, the buffer is most efficient at most efficient at pHpH where the first derivative has a minimum where the first derivative has a minimum or the or the second derivative is zerosecond derivative is zero..


Find where an unfamiliar function Find where an unfamiliar function ff((xx) crosses axis:) crosses axis:

Useful rekomendationsUseful rekomendations

a value of a value of ff((xx) at ) at x x = 0;= 0;

a value of a value of xx at which at which ff((xx)) = 0;= 0;

Find location of stacionary pointsFind location of stacionary points(i.e. (i.e. xx value(s) where the first derivative value(s) where the first derivative f f ''((xx)=0 ))=0 )

Check the sign of the second derivative at Check the sign of the second derivative at xx values of the stacionary points values of the stacionary points(i.e. determine whether stacionary point is maximum or minimum(i.e. determine whether stacionary point is maximum or minimum ) )

Find the limiting behaviour of the functionFind the limiting behaviour of the function(i.e. when values of (i.e. when values of xx are extremely large) are extremely large)

Differential calculus

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