Partial derivatives
2
2
1. Let f (x, y) = e(x +y ) + x2 + y 2 + xy + 2y + 3.
∂f
∂f
a) Compute
and
.
∂x
∂y
b) Show the second partials can be computed in any order. That is,
∂2f
∂2f
=
.
∂x∂y
∂y∂x
∂f
(1, 3).
∂x
∂f
2
2
Answer: a)
= 2xe(x +y ) + 2x + y,
∂x
c) Find
∂f
2
2
= 2ye(x +y ) + 2y + x + 2.
∂y
∂f
∂2f
we compute the partial with respect to x of
.
∂x∂y
∂y
� �
�
∂2f
∂ ∂f
∂ � (x2 +y2 )
2
2
=
=
2ye
+ 2y + x + 2 = 4xye(x +y ) + 1.
∂x∂y
∂x ∂y
∂x
b) To compute
Likewise
∂2f
∂
=
∂y∂x
∂y
�
∂f
∂x
�
=
�
∂ �
2
2
2
2
2xe(x +y ) + 2x + y = 4xye(x +y ) + 1.
∂y
We have shown the order of differentiation didn’t matter.
∂f
c) Evaluating
(1, 3) = 2e10 + 5.
∂x
MIT OpenCourseWare
http://ocw.mit.edu
18.02SC Multivariable Calculus
Fall 2010
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