Thermodynamics and Chemistry

Appendix E Calculus Review

Appendix E Calculus Review

E.1 Derivatives

Letfbe a function of the variablex, and letÅf be the change inf whenxchanges by
Åx. Then thederivativedf=dxis the ratioÅf=Åxin the limit asÅxapproaches zero.
The derivative df=dxcan also be described as the rate at whichf changes withx, and as
the slope of a curve offplotted as a function ofx.
The following is a short list of formulas likely to be needed. In these formulas,uandv
are arbitrary functions ofx, andais a constant.

d.ua/

dx

Daua^1

du

dx

d.uv/

dx

Du

dv

dx

Cv

du

dx

d.u=v/

dx

D



1

v^2



v

du

dx

u

dv

dx



d ln.ax/

dx

D

1

x

d.eax/

dx

Daeax

df .u/

dx

D

df .u/

du

du

dx

E.2 Partial Derivatives

Iffis a function of the independent variablesx,y, andz, thepartial derivative.@f=@x/y;z
is the derivative df=dxwithyandzheld constant. It is important in thermodynamics
to indicate the variables that are held constant, as.@f=@x/y;zis not necessarily equal to
.@f=@x/a;bwhereaandbare variables different fromyandz.
The variables shown at the bottom of a partial derivative should tell you which vari-
ables are being used as the independent variables. For example, if the partial derivative is
@f
@y



a;b

thenfis being treated as a function ofy,a, andb.

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