James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

SectION 14.3 Partial Derivatives 913
By taking h − 5 and 25, we approximate G9s70d using the tabular values:
G9s70d <
Gs75d 2 Gs70d
5
−
f s96, 75d 2 f s96, 70d
5
−
130 2 125
5
− 1
G9s70d <
Gs65d 2 Gs70d
25
−
f s96, 65d 2 f s96, 70d
25
−
121 2 125
25
− 0.8
By averaging these values we get the estimate G9s70d < 0.9. This says that, when the
temperature is 968F and the relative humidity is 70%, the heat index rises about 0.98F for
every percent that the relative humidity rises.
In general, if f is a function of two variables x and y, suppose we let only x vary while
keeping y fixed, say y − b, where b is a constant. Then we are really considering a function
of a single variable x, namely, tsxd − f sx, bd. If t has a derivative at a, then we call
it the partial derivative of f with respect to x at sa, bd and denote it by f x sa, bd. Thus
1 f x sa, bd − t9sad where tsxd − f sx, bd
By the definition of a derivative, we have
and so Equation 1 becomes
t9sad − lim
h l 0
tsa 1 hd 2 tsad
h
2 f x sa, bd − lim
h l 0
f sa 1 h, bd 2 f sa, bd
h
Similarly, the partial derivative of f with respect to y at sa, bd, denoted by f y sa, bd, is
obtained by keeping x fixed sx − ad and finding the ordinary derivative at b of the function
Gsyd − f sa, yd:
3 f y sa, bd − lim
h l 0
f sa, b 1 hd 2 f sa, bd
h
With this notation for partial derivatives, we can write the rates of change of the heat
index I with respect to the actual temperature T and relative humidity H when T − 968F
and H − 70% as follows:
f T s96, 70d < 3.75 f H s96, 70d < 0.9
If we now let the point sa, bd vary in Equations 2 and 3, f x and f y become functions of
two variables.
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