The Manga Guide to Calculus

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

Derivatives of Implicit Functions

A point (x, y) for which a two-variable function f(x, y) is equal to constant c

describes a graph given by f(x, y) = c. When a part of the graph is viewed as a

single-variable function y = h(x), it is called an implicit function. An implicit

function h(x) satisfies f(x, h(x)) = c for all x defined. We are going to obtain

h(x) here.

When z = f(x, y), the formula of total differentials is written as dz = f

x

dx +

f

y

dy. If (x, y) moves on the graph of f(x, y) = c, the value of the function f(x, y)

does not change, and the increment of z is 0, that is, dz = 0 ...