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Now
consider a function, f(x,y), which has two independent variables x
and y.
Strategy: Extend
the definition of the limit for a function of one independent
variable
to a function of two independent variables leading to the partial
derivatives.
(Nonrigorous approach given below)
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The
partial derivative of f(x,y) with respect to x
(holding y
constant) is:
fx(x,y) =
lim [ f(x + h, y) -
f(x,y)] / h
h
→ 0
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The
partial derivative of f(x,y) with respect to y (holding x constant) is:
fy(x,y) =
lim [ f(x, y + k) -
f(x,y)] / k
k →
0
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With a function,
f(x,y), of two variables fx(x,y)
represents the slope of f(x,y) in
the
x-direction whereas fy(x,y) represents the slope of f(x,y) in the y-direction.
Notation: fx(x,y) = ∂f(x,y)/
∂x and fy(x,y) = ∂f(x,y)/
∂y
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