Fermat's Library | Geometrical Meaning of Differentiation annotated/explained version.

### TL;DR This is a text about differentiation and its geometric...

You can find the entire book here: - [Calculus Made Easy (HTML v...

Silvanus Phillips Thompson was a professor of physics mostly known ...

A visual example of varying slopes at different points of a curve. ...

The tangent to the curve at specific point is parametrized by a lin...

That's funny

Values for the tangent function $\tan$ between $-2\pi$ and $2\pi$. ...

ahhhh

local minimum

In this figure we can see one maximum at $x=1$ and one minimum at $...

The simplest case to examine is that of a linear function. The slop...

This is a great example where you can visualize the graph for a giv...

let's talk about this

CHAPTER X.

GEOMETRICAL MEANING OF DIFFERENTIATION.

It is useful to consider what geometrical meaning can be given to the

di↵erential coeﬃcient.

In the ﬁrst place, any function of x,such,forexample,asx

,or

or ax + b,canbeplottedasacurve;andnowadayseveryschoolboyis

familiar with the process of curve-plotting.

O X

Fig. 7.

Let PQR,inFig. 7,beaportionofacurveplottedwithrespect

to the axes of coordinates OX and OY .ConsideranypointQ on this

curve, where the abscissa of the point is x and its ordinate is y. Now

observe how y changes w hen x is varied. If x is made to incr ea se by

CALCULUS MADE EASY 76

asmallincrementdx,totheright,itwillbeobservedthaty also (in

this particular curve) increases by a small increment dy (because this

particular curve happens to be an ascending curve). Then the ratio of

dy to dx is a measure of the degree to which the curve is sloping up

between the two points Q and T . As a matter of fact, it can be seen on

the ﬁgure that the curve between Q and T has m a ny di↵erent slopes,

so that we cann ot very well speak of the slope of the curve between

Q and T .If,however,Q and T are so near each other that the small

portion QT of the curve is practically straight, then it is true to say that

the ratio

is the slope of the curve along QT .ThestraightlineQT

produced on either side touches the curve along the portion QT only,

and if this portio n is indeﬁnitely small, the straight line will touch the

curve at practically one point only, and be therefore a tangent to the

curve.

This tangent to th e curve has evidently the same slope as QT ,so

that

is the slope of the tangent to the curve at the point Q for which

the value of

is found.

We have seen that the short expression “the slope of a curve” has

no precise meaning, because a curve has so many slopes—in fact, every

small portion of a curve has a di↵erent slope. “The slope of a curve at

a point”is,however,aperfectlydeﬁnedthing;itistheslopeofavery

small portion of the curve situated just at that point; and we have seen

that this is the same as “the slope of the tan gent to the curve at that

point.”

Observe that dx is a short step to the right, and dy the correspond-

ing short step upwards. These steps must be considered as short as

MEANING OF DIFF ER ENTI ATION 77

possible—in fact indeﬁnitely short,—though in diagrams we have to

represent them by bits that are not inﬁni t es i mal l y small, otherwise

they could not be seen.

We shall hereafter make considerable use of this circumstance that

represents the slope of the curve at any point.

O X

Fig. 8.

If a curve is sloping up at 45



at a particular point, as in Fig. 8, dy

and dx will be equal, and the value of

=1.

If the curve slopes up steeper t h an 45



(Fig. 9),

will be greater

than 1.

If the curve slopes up very gently, as i n Fig. 10,

will be a fraction

smaller than 1.

For a horizontal line, or a horizontal place in a curve, dy =0,and

therefore

=0.

If a curve slopes downward,asinFig. 11, dy will be a step down,

and must t h er efo re be reckoned of negative va l u e; hence

will have

negative sign also.

CALCULUS MADE EASY 78

O X

Fig. 9.

O X

Fig. 10.

If the “curve” happens to be a straight line, like th at in Fig. 12,the

value of

will be the same at all points along i t . In other words its

slope is constant.

If a cur ve is one that turn s mor e u pwards as it goes along to the

right, the values of

will b ecom e greater and greater with th e in-

creasing steepness, as in Fig. 13.

If a curve is one that gets ﬂatter and ﬂatter as it goes along, the

values of

will become smaller and smaller as the ﬂatter par t is

O X

Fig. 11.

MEANING OF DIFF ER ENTI ATION 79

O X

Fig. 12.

O X

Fig. 13.

reached, as in Fig. 14.

If a curve ﬁrst descends, and then goes u p agai n , as in Fig. 15,

presenting a concavity upwards, then clearly

will ﬁrst be negative,

with diminishing values as the curve ﬂattens, then will be zero at the

point where the bottom of the trough of the curve is reached; and from

this point onward

will have positive values that go on incr easin g. In

such a case y is said to pass by a minimum.Theminimumvalueofy is

not necessarily the smallest value of y,itisthatvalueofy corresponding

to the bottom of the trough; for instance, in Fig. 28 (p. 99), the value

of y corr espondi n g to the bottom of the trough is 1, while y takes

CALCULUS MADE EASY 80

O X

Fig. 14.

O X

y min.

Fig. 15.

elsewhere values which are smaller than this. The characteristic of a

minimum is that y must increase on either side of it.

N.B.—For the particular value of x that makes y a minimum,the

value of

=0.

If a curve ﬁrst ascends and then descends, the values of

will be

positive at ﬁrst; then zero, as the summit is reached; then negative,

as the curve slopes downwards, as in Fig . 16.Inthiscasey is said to

pass by a maximum,butthemaximumvalueofy is not necessarily the

greatest value of y.InFig. 28,themaximumofy is 2

,butthisis

by no means the greatest value y can have at some other point of the

curve.

N.B.—For the particular value of x that makes y a maximum,the

value of

=0.

If a curve has the peculiar form of Fig. 17,thevaluesof

will

always be positive; but there will be one particular place where t h e

slope is least steep, where the value of

will be a minimum; that is,

less than it is at any other part of the curve.

MEANING OF DIFF ER ENTI ATION 81

O X

y max.

Fig. 16.

O X

Fig. 17.

If a curve has the form of Fi g . 18,thevalueof

will be negative

in the upper part, and positive in the lower part; while at the nose of

the curve where it becomes actually perpendicular, the value of

will

be inﬁnitely great.

O X

Fig. 18.

Now that we understand that

measures the steepness of a curve

at any point, let us turn to some of the equations which we have already

learned how to di↵erentiate.

CALCULUS MADE EASY 82

(1) As the simplest case take this:

y = x + b.

It is plotted out in Fig. 19,usingequalscalesforx and y.Ifweput

x =0,thenthecorrespondingordinatewillbey = b;thatistosay,the

“curve” crosses the y-axi s at the height b.Fromhereitascendsat45



;

O X

Fig. 19.

O X

Fig. 20.

for whatever values we give to x to the right, we have an equal y to

ascend. The line has a gradient of 1 in 1.

Now di↵erentiate y = x + b,bytheruleswehavealreadylearned

(pp. 21 and 25 ante), and we get

=1.

The slope of the l i n e is such t h at for every little step dx to the right,

we go an equal little step dy upward. And this slope is constant—always

the same slope.

(2) Take another case:

y = ax + b.

MEANING OF DIFF ER ENTI ATION 83

We know that this curve, like the preceding one, will start from a

height b on the y-axis. But before we draw the curve, let us ﬁnd i ts

slope by di↵erentiating; which gives

= a.Theslopewillbeconstant,

at an angle, the tangent of which is here called a.Letusassigntoa

some numerical value—say

. Then we must give it such a slope that

it ascends 1 in 3; or dx will be 3 times as great as dy;asmagniﬁedin

Fig. 21.So,drawthelineinFi g. 20 at this slope.

Fig. 21.

(3) Now for a slightly harder case.

Let y = ax

+ b.

Again the curve will start on the y-axis at a height b above the

origin.

Now di↵erentiat e. [If you have forgotte n , turn back to p. 25;or,

rather, don’t turn back, but think out the di↵erentiation.]

=2ax.

This shows that the steepness will not be constant: it increases as

x increases. At the starting point P ,wherex =0,thecurve(Fig. 22)

has no steepness—that is, it is level. On the left o f the origin, where x

has negative values,

will also have negative valu es, or will descend

from left to right, as in the Figure.

CALCULUS MADE EASY 84

O X

Fig. 22.

Let us illustrate this by working out a particular instance. Taking

the equation

y =

+3,

and di↵erentiating it, we get

Now assign a few successive values, say from 0 to 5, to x;andcalculate

the corresponding values of y by the ﬁrst equation; and of

from the

second equat i on . Tab u l at i ng results, we have:

x 0 1 2 3 4 5

y 3 3

4 5

7 9

1 1

2 2

Then plot them out in two curves, Figs. 23 and 24,inFig. 23 plotting

the values of y against those of x and in Fig. 24 those of

against

those of x.Foranyassignedvalueofx,theheight of the ordinate in

the second curve is proportional to the slope of the ﬁrst curve.

MEANING OF DIFF ER ENTI ATION 85

3 2 10 1 2 3 4 5

y =

Fig. 23.

3 2 1

12345

Fig. 24.

If a curve comes to a sudden cusp, as in Fig. 25,theslopeatthat

point suddenly changes from a slope upward to a slope downward. In

O X

Fig. 25.

that case

will clearly undergo an abrupt change from a positive to

anegativevalue.

The following examples show further applications of the prin cip l es

just explain ed .

CALCULUS MADE EASY 86

(4) Find the slope of the tangent to the curve

y =

+3,

at the point where x = 1. Find the angle which this tangent makes

with the curve y =2x

+2.

The slope of the tangent is the slope of t h e curve at the point where

they touch one another (see p. 76); that is, it is the

of the curve for

that point. Here

= 

and for x = 1,

= 

,whichisthe

slope of the tangent and of the curve at that point. The tangent, being

astraightline,hasforequationy = ax + b,anditsslopeis

= a,

hence a = 

. Also if x = 1, y =

2(1)

+3 = 2

;andasthe

tangent passes by this point, the coordinates of the point must satisfy

the equation of the tangent, namely

y = 

x + b,

so that 2

= 

⇥ (1) + b and b = 2; the eq u a t i on of the tangent is

therefore y = 

x +2.

Now, when two curves meet, the intersection being a point com-

mon to both curves, its co ordinates must satisfy the equation of each

one of the two curves; that is, it must be a solution of the system of

simultaneous equations formed by coupling together the equations of

the curves. Here the curves meet o n e another at points given by the

solution of

y =2x

+2,

y = 

x +2 or 2x

+2=

x +2;

MEANING OF DIFF ER ENTI ATION 87

that is, x(2x +

)=0.

This equation has for its solutions x =0andx = 

.Theslopeof

the curve y =2x

+2atanypointis

=4x.

For the point where x = 0, this slope is zero; the curve is h or i zo ntal.

For the p oint where

x = 

= 1;

hence the curve at that point slopes downwards to the right at such

an angle ✓ with the horizontal that tan ✓ = 1; that is, at 45



to the

horizontal.

The slope of the straight line is 

;thatis,itslopesdownwardsto

the right and makes with the horizontal an angle  such that t an  =

;

that is, an angle of 26



.Itfollowsthatattheﬁrstpointthecurve

cuts the straight line at an angle of 26



,whileattheseconditcuts

it at an angle of 45



 26



=18



(5) A straight line is to be drawn, through a point whose coordinates

are x =2,y = 1, as tangent to the curve y = x

 5x + 6. Find the

coordinates of the p oi nt of contact.

The sl ope of the tangent must be the same as the

of the curve;

that is, 2x  5.

The equation of the straight line is y = ax+b,andasitissatisﬁedfor

the values x =2,y = 1, then 1=a⇥2+b;also,its

= a =2x5.

The x and the y of the point of contact must also satisfy both the

equation o f the tangent and the equation of the curve.

CALCULUS MADE EASY 88

We have then

y = x

 5x +6, (i)

y = ax + b, (ii)

1=2a + b, (iii)

a =2x  5, (iv)

four equation s in a, b, x, y.

Equations (i ) and (ii) give x

 5x +6=ax + b.

Replacing a and b by their v alue in this, we get

 5x +6=(2x  5)x  1  2(2x  5),

which simpli ﬁ es to x

 4x +3=0,thesolutionsofwhich are: x =3

and x =1. Replacingin(i),wegety =0andy =2respectively;the

two points of contact are then x =1,y =2,andx =3,y =0.

Note.—In all exercises dealing with curves, students will ﬁnd it ex-

tremely instructive to verify the deductions obtained by actually plot-

ting the curves.

Exercises VII I . (See page 256 for Answers.)

(1) Plot the c urve y =

5, using a scale of millimetres. Measure

at points corresponding to di↵erent values of x,theangleofitsslope.

Find, by di↵erentiating the equation, the expression for slope; and

see, from a Table of Natural Tangents, whether this agrees with the

measured angle.

MEANING OF DIFF ER ENTI ATION 89

(2) Find what will be the slope of the curve

y =0.12x

 2,

at the particular point that has as abscissa x =2.

(3) If y =(x  a)(x  b) , show that at the particular point of the

curve wh er e

=0,x will have the value

(a + b).

(4) Find the

of the equation y = x

+3x;andcalculatethe

numerical values of

for the points corresponding to x =0,x =

x =1,x =2.

(5) In the curve to which the equation is x

=4,ﬁndthevalues

of x at those points where the slope = 1.

(6) Find the slope, at any point, of the curve whose equation is

=1;andgivethenumericalvalueoftheslopeattheplace

where x =0,andatthatwherex =1.

(7) The equation of a tangent to the curve y =52x +0.5x

,being

of the form y = mx + n,wherem and n are constants , ﬁnd the value

of m and n if the point where the tangent touches the curve has x =2

for abscissa.

(8) At what angle do the two curves

y =3.5x

+2 and y = x

 5x +9.5

cut one another?

(9) Tangents to the curve y = ±

25  x

are drawn at points for

which x =3andx = 4. Fin d the coordinates of the point of intersection

of the tangents and their mutual inclination.

CALCULUS MADE EASY 90

(10) Astraightliney =2x  b touches a curve y =3x

+2at one

point. What are the coordinat es of the point of contact, and what is

the value of b?

Discussion

This is a great example where you can visualize the graph for a given function $y=f(x)$ on Figure 23 and the graph for it's derivative $\frac{dy}{dx}=f^{\prime}(x)$ of that function Figure 24. The simplest case to examine is that of a linear function. The slope $a$ of a linear function is a constant. In this first example the author choose the easiest of all $a=1$. The tangent to the curve at specific point is parametrized by a linear function as follows $y = ax+b$ where a is the slope of that curve and is computed as follows: \begin{eqnarray*} a&=&{\frac {{\text{change in }}y}{{\text{change in }}x}}\\ &=&{\frac {\Delta y}{\Delta x}}\\ &=&{\tan{\theta}} \end{eqnarray*} ![differentiation](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Wiki_slope_in_2d.svg/562px-Wiki_slope_in_2d.svg.png) *Figure: Slope of a linear function.* local minimum Values for the tangent function $\tan$ between $-2\pi$ and $2\pi$. Note that for an angle $\theta = 45^{\circ} = \frac{\pi}{4}$: $\tan{(45^{\circ})}= \tan{(\frac{\pi}{4})} = 1$ !["tan plot"](https://i.imgur.com/BoRR8gT.png) A visual example of varying slopes at different points of a curve. Here the values for the slope are given by $f^{\prime}$ at different points $A$ for the function $f(x)=x\sin{(x^2)}+1$. !["slopes"](https://upload.wikimedia.org/wikipedia/commons/2/2d/Tangent_function_animation.gif) let's talk about this ahhhh That's funny In this figure we can see one maximum at $x=1$ and one minimum at $x=3$. The slope **a** of the tangent to the curve at those points is 0, i.e. $a=\frac{dy}{dx} = 0$. To the left of a minimum we have that $a<0$ and to the right $a>0$. Inversely for a maximum we have that to the left of it the slope is $a>0$ and to the right $a<0$. !["figure convex"](https://i.imgur.com/ExouW1t.png) Silvanus Phillips Thompson was a professor of physics mostly known for his work as an electrical engineer. He was elected to the Royal Society 1891. Thompson was also an author and he published his most famous book **Calculus Made Easy** in 1910. This book is one of the greatest introductions to the fundamentals of infinitesimal calculus, and is still in print today. Learn more about the author here: - [Silvanus P. Thompson 1851-1916 Professor of Physics - Electrical Engineer](http://www.engineerswalk.co.uk/st_walk.html) - [Silvanus P. Thompson](https://en.wikipedia.org/wiki/Silvanus_P._Thompson) !["spt"](https://upload.wikimedia.org/wikipedia/commons/4/4a/Thompson_Silvanus_mature.jpg) ### TL;DR This is a text about differentiation and its geometrical meaning from the book "Calculus Made Easy" originally published in 1910 by Silvanus P. Thompson. It constitutes one of the greatest and most elegant introductions to the topic of calculus. It's worth reading even if you know math really well because it will help you understand how to teach complexity without any added fluff or overt technicality to others. It's a fascinating book. Excerpt from its Prologue: > ***Considering how many fools can calculate, it is surprising that it should be thought either a difficult or a tedious task for any other fool to learn how to master the same tricks.*** >*** Some calculus tricks are quite easy. Some are enormously difficult. The fools who write the textbooks of advanced mathematics - and they are mostly clever fools - seldom take the trouble to show you how easy the easy calculations are. On the contrary, they seem to desire to impress you with their tremendous cleverness by going about it in the most difficult way.*** >*** Being myself a remarkably stupid fellow, I have had to unteach myself the difficulties, and now beg to present to my fellow fools the parts that are not hard. Master these thoroughly, and the rest will follow. What one fool can do, another can.*** You can find the entire book here: - [Calculus Made Easy (HTML version)](https://calculusmadeeasy.org) - [The Project Gutenberg EBook of Calculus Made Easy, by Silvanus Thompson (PDF version)](http://www.gutenberg.org/files/33283/33283-pdf.pdf)

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