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PartialDerivatives 6.2 OBJECTIVE
• Find the partial derivatives of a given function.
• Evaluate partial derivatives.
• Find the four second-order partial derivatives of a function in two variables.
2012 Pearson Education, Inc.
All rights reserved Slide 6.2-2
6.2PartialDerivatives
DEFINITION:
For z = f (x, y), the partial derivatives with respect to x and y are
and
¶z ¶x ¶z ¶y
= lim h®0
= lim h®0
f(x + h,y)
h f(x,y+ k)
k
f(x,y),
f(x,y),
2012 Pearson Education, Inc. All rights reserved Slide 6.2-3
6.2PartialDerivatives
Example 1: For w = x2 xy+ y2 + 2yz+ 2z2 + z,
find
¶w ¶w ¶x ¶y
and
¶w ¶z
In order to find ¶w ¶x, we regard x as the variable (biến) and y and z as constants (hằng số)
¶w ¶x
= 2x y
2012 Pearson Education, Inc. All rights reserved Slide 6.2-4
6.2PartialDerivatives
Example 1 (concluded):
Similarly, from w = x2 xy+ y2 + 2yz+ 2z2 + z, we get
w
y
= x+ 2y+ 2z
and
w
z
= 2y + 4z +1
2012 Pearson Education, Inc. All rights reserved Slide 6.2-5
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