218 Chapter 6 Let’s Learn About Partial Dierentiation!
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
When z = f(x, y), the formula of total differentials is written as dz = f
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. Then, we get
0 = f
dx + f
dy. Assuming f
≠ 0 and modifying this, we get
The left side of this equation is the ideal expression of the increment
of y divided by the increment of x at a point on the graph. It is exactly the
derivative of h(x). Thus,
= −h x
f(x, y) = r
, where f(x, y) = x
, describes a circle of radius r centered
at the origin. Near a point that satisfies x
, we can solve f(x, y) = x
to find the implicit function y = h(x) = r
y h x r x=
= − −
Then, from the formula, the derivative of these functions is given by
= − = −h x
1. Obtain f
for f(x, y) = x
+ 2xy + 3y
2. Under the gravitational acceleration g, the period T of a pendulum hav-
ing length L is given by
(the gravitational acceleration g is known to vary depending on the
height from the ground).
Obtain the expression for total differential of T.
If L is elongated by 1 percent and g decreases by 2 percent, about
what percentage does T increase?
3. Using the chain rule, calculate the differential formula of the implicit
function h(x) of f(x, y) = c in a different way than above.
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What Is Mathematics For? 221
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