Page 38 - Physical Chemistry

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Figure 1.10 shows that the derivative of the function y f(x) at a given point is equal Section 1.6
to the slope of the curve of y versus x at that point. Differential Calculus
2
As a simple example, let y x . Then
2
1x h2 x 2 2xh h 2
f¿1x2 lim lim lim 12x h2 2x
hS0 h hS0 h hS0
2
The derivative of x is 2x.
A function that has a sudden jump in value at a certain point is said to be discon-
tinuous at that point. An example is shown in Fig. 1.11a. Consider the function y
x , whose graph is shown in Fig. 1.11b. This function has no jumps in value anywhere
and so is everywhere continuous. However, the slope of the curve changes suddenly
at x 0. Therefore, the derivative y is discontinuous at this point; for negative x the
function y equals x and y equals 1, whereas for positive x the function y equals x
and y equals 1. Figure 1.10
Since f (x) is defined as the limit of y/ x as x goes to zero, we know that, for
small changes in x and y, the derivative f (x) will be approximately equal to y/ x. As point 2 approaches point 1, the
quantity y/ x tan u approaches
Thus y f (x) x for x small. This equation becomes more and more accurate as the slope of the tangent to the
x gets smaller. We can conceive of an infinitesimally small change in x, which we curve at point 1.
symbolize by dx. Denoting the corresponding infinitesimally small change in y by dy,
we have dy f (x) dx, or

dy y¿1x2 dx (1.26)*
The quantities dy and dx are called differentials. Equation (1.26) gives the alternative
notation dy/dx for a derivative. Actually, the rigorous mathematical definition of dx
and dy does not require these quantities to be infinitesimally small; instead they can
be of any magnitude. (See any calculus text.) However, in our applications of calculus
to thermodynamics, we shall always conceive of dy and dx as infinitesimal changes.
Let a and n be constants, and let u and v be functions of x; u u(x) and v v(x).
Using the definition (1.25), one finds the following derivatives:
n
ax
da d1au2 du d1x 2 d1e 2
0, a , nx n 1 , ae ax
dx dx dx dx dx
d ln ax 1 d sin ax d cos ax
, a cos ax, a sin ax
dx x dx dx
(1.27)*
d1u v2 du dv d1uv2 dv du
, u v
dx dx dx dx dx dx
1
d1u>v2 d1uv 2 dv du
uv 2 v 1
dx dx dx dx
The chain rule is often used to find derivatives. Let z be a function of x, where x
is a function of r; z z(x), where x x(r). Then z can be expressed as a function of r;
z z(x) z[x(r)] g(r), where g is some function. The chain rule states that dz/dr
(dz/dx)(dx/dr). For example, suppose we want (d/dr) sin 3r . Let z sin x and x
2
2
2
2
3r . Then z sin 3r , and the chain rule gives dz/dr (cos x) (6r) 6r cos 3r .
Equations (1.26) and (1.27) give the following formulas for differentials:
n
ax
ax
d1x 2 nx n 1 dx, d1e 2 ae dx
(1.28)*
d1au2 a du, d1u v2 du dv, d1uv2 u dv v du
Figure 1.11
We often want to find a maximum or minimum of some function y(x). For a (a) A discontinuous function.
function with a continuous derivative, the slope of the curve is zero at a maximum or (b) The function y x .