The Manga Guide to Calculus

Partial Dierentiation 195

Similarly, for y we get

fabtfabfab t

y

,,,+

()

−

()

≈

()

ββ

Substituting this in ,

≈+

()

+

()

−

()

−

()

=+

()

−

()

fabt fabfab fab

fabtfab

xy

xy

xx

,,

,,

,,

βα βαβ

β

(()

α

 

Since f

x

(a, b +

β 

t) − f

x

(a, b) ≈ 0 if t is close enough to 0, the relative error =

 ≈ 0. Thus, we have shown “the relative error

→

0 when AP

→

0 in any

direction.”

It should be noted that f

x

must be continuous to say f

x

(a, b +

β 

t) − f

x

(a, b)

≈ 0 (t ≈ 0). Unless it is continuous, we don’t know whether the derivative

exists in every direction, even though f

x

and f

y

exist. Since such functions

are rather exceptional, however, we won’t cover them in this book.

Examples (Functio