bj f(x,y)= еху cЈ fexiy>= x cosx cобу p² d.j fexiys= (x'y 2) log (x'ty ?).

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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B,c,d
Etol villa
20
morenability for Functions of Two Variables
Using the linear approximation, we are ready to define the notion of differentiability.
Definition Differentiable: Two Variables Let f: R² → R. We say f is
differentiable at (xo, yo), if af/ax and af/ay exist at (xo, yo) and if
f(x, y)-f(xo, yo) -
[af
əx
f(xo, yo) +
-(xo, yo) (x-xo) -
||(x, y) - (xo, yo) ||
[af (x -
-(xo, yo)
əx
- x0) +
af
as (x,y) → (xo, yo). This equation expresses what we mean by saying that
[af
Jo) ] (Y - YO)
yo)
ду
is a good approximation to the function f.
-(xo, yo) (y-yo)
Əy
-0
-(xo, yo)
(2)
polecon
bour ow isdi tuo a
It is not always easy to use this definition to see whether f is differentiable, but it
will be easy to use another criterion, given shortly in Theorem 9.
Transcribed Image Text:Etol villa 20 morenability for Functions of Two Variables Using the linear approximation, we are ready to define the notion of differentiability. Definition Differentiable: Two Variables Let f: R² → R. We say f is differentiable at (xo, yo), if af/ax and af/ay exist at (xo, yo) and if f(x, y)-f(xo, yo) - [af əx f(xo, yo) + -(xo, yo) (x-xo) - ||(x, y) - (xo, yo) || [af (x - -(xo, yo) əx - x0) + af as (x,y) → (xo, yo). This equation expresses what we mean by saying that [af Jo) ] (Y - YO) yo) ду is a good approximation to the function f. -(xo, yo) (y-yo) Əy -0 -(xo, yo) (2) polecon bour ow isdi tuo a It is not always easy to use this definition to see whether f is differentiable, but it will be easy to use another criterion, given shortly in Theorem 9.
Find af lax, af/ dy if
a
a) f(x,y) = xy
bi f(x,y)= exy
c. f(xiy) = x COSX cosy
p=
lag
d.) f(x,y)= (x²+ y²) log (x²+y²).
Transcribed Image Text:Find af lax, af/ dy if a a) f(x,y) = xy bi f(x,y)= exy c. f(xiy) = x COSX cosy p= lag d.) f(x,y)= (x²+ y²) log (x²+y²).
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