The Manga Guide to Calculus

198 Chapter 6 Let’s Learn About Partial Dierentiation!

Since

∂z

—

∂x

Δx is “the increment of z in the x direction when y is fixed at b”

and

∂z

—

∂y

Δy is “the increment in the y direction when x is fixed at a,” expres-

sion { also means “the increment of z = f(x, y) is the sum of the increment

in the x direction and that in the y direction.”

When expression { is idealized (made instantaneous), we have

|

dz

z

x

dx

z

y

dy=

∂

∂

+

∂

∂

or

}

df f dx f dy

x y

= +

(Δ has been changed to d.)

The meaning of the formula is as follows.

Increment of height of a curved surface =

Increment in

the y direction

Partial derivative

in the x direction

×

Increment in

the x direction

+ ...