Multivariate Calculus - Bruce E. Shapiro

Lecture 9
The Partial Derivative
Definition 9.1 Let f(x, y) be a function of two variables.
derivative of f with respect to x is defined as
Then the partial
∂f
∂x = f f(x + ∆x, y) − f(x, y)
x(x, y) = lim
∆x→0 ∆x
(9.1)
if the limit exists, and the the partial derivative of f with respect to y is
similarly defined as
∂f
∂y = f f(x, y + ∆y) − f(x, y)
y(x, y) = lim
∆y→0 ∆y
(9.2)
if that limit exists.
Calculation of partial derivatives is similar to calculation of ordinary derivatives.
To calculate ∂f/∂x, for example, differentiate with respect to x while treating y as
a constant; to calculate ∂f/∂y, differentiate with respect to y while treating x as a
constant.
Example 9.1 Find the partial derivatives of f(x, y) = x.
Solution. Applying equations (9.1) and (9.2), we calculate
∂f
∂x = f f(x + h, y) − f(x, y)
x(x, y) = lim
h→0 h
∂f
∂y
(x + h) − x
= lim
= 1
h→0 h
f(x, y + h) − f(x, y)
= f y (x, y) = lim
h→0 h
x − x
= lim = 0.
h→0 h
All of the usual rules of differentiation apply to partial derivatives, including
things like the product rule, the quotient rule, and the derivatives of basic functions.
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