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.