phy1020.DVI

Cancelingxin the numerator and denominator,

slopeD lim

x! 0

2xCx (4.15)

and asxapproaches zero,

slopeD2x (4.16)

So for at any point along the curvef.x/Dx^2 , its slope is given by2x.AtxD 3 , the slope is 2  3 D 6 ,in
agreement with our earlier approximations.
The slope is called thederivativeoff.x/with respect tox. As we have just shown, the derivative of
f.x/Dx^2 with respect toxis2x. We indicate the derivative ofyDf.x/with respect toxby the notation

dy

dx

or

d

dx

f.x/ (4.17)

Thus the derivative can be thought of as the quotient of two infinitesimal numbers, and is defined as

dy

dx

 lim

x! 0

y

x

D lim

x! 0

f.xCx/f.x/

x

(4.18)

For our exampleyDf.x/Dx^2 ,

dy

dx

D

d

dx

x^2 D2x (4.19)

More generally, it can be shown that for anyn,

d

dx

xnDnxn^1 (4.20)

For example,

d

dx

x^5 D5x^4 (4.21)

Herenneed not necessarily be an integer. For example, since

p

xDx1=2,wehave

d

dx

p

xD

d

dx

x1=2D

1

2

x1=2D

1

2

p

x

(4.22)

Similar results can be worked out for many common functions. Appendix E gives a short table of deriva-
tives. In conjunction with this table, we note the following properties (uandvare functions ofx, andais a
constant):

d

dx

.au/Da

du

dx

(4.23)

d

dx

.uCv/D

du

dx

C

dv

dx

(4.24)

d

dx

.uv/D

du

dx



dv

dx

(4.25)

d

dx

.uv/D

du

dx

vCu

dv

dx

(4.26)

d

dx

u

v



D

v.du=dx/u.dv=dx/

v^2

(4.27)