Multivariate Calculus - Bruce E. Shapiro

60 LECTURE 9. THE PARTIAL DERIVATIVE
Example 9.2 Find the partial derivatives with respect to x and y of f(x, y) =
e y sin x.
Solution.
∂f
∂x = ∂
∂x (ey sin x) = e y ∂
∂x (sin x) = ey cos x (9.3)
∂f
= ∂ ∂y ∂y (ey sin x) = sin x ∂ ∂y (ey ) = e y sin x (9.4)
Example 9.3 Find the partial derivatives of f(x, y) = x 3 y 2 − 5x + 7y 3 .
Solution.
∂f
∂x = ∂
∂x (x3 y 2 − 5x + 7y 3 )
∂f
∂y
= ∂
∂x (x3 y 2 ) − ∂
∂x (5x) + ∂
∂x (7y3 )
= y 2 ∂
∂x (x3 ) − ∂
∂x (5x) + ∂
∂x (7y3 )
= 3x 2 y 2 − 5
= ∂ ∂y (x3 y 2 − 5x + 7y 3 )
= ∂ ∂y (x3 y 2 ) − ∂ ∂y (5x) + ∂ ∂y (7y3 )
= x 3 ∂ ∂y (y2 ) − ∂ ∂y (5x) + ∂ ∂y (7y3 )
= 2x 3 y + 21y 2
Physical Meaning of Partials.
Partial derivatives represent rates of change, just as ordinary derivatives. If T (x,t)
is the temperature of an object as a function of position and time then
∂T
(x, t)
∂x
represents the change in temperature at time t with respect to position, i.e., the
slope of the temperature vs. position curve, and
∂T
(x, t)
∂t
gives the change in temperature at a fixed position x, with respect to time.
If, on the other hand, T (x,y) gives the temperature of an object as a function
of position in the xy plane then
∂T
(x, y)
∂x
Revised December 6, 2006. Math 250, Fall 2006