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UNIVERSITY OF TECHNOLOGYMATERIAL ENGINEERING
DEPARTMENT
MATHEMATICS II
DR. KADHUM MUTTAR SHABEEB
First Semester 2011-2012
Partial Derivatives
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Functions of Several Variables
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Domains and Ranges
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Limits and Continuity
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Partial Derivatives of a Function of Two Variables
PARTIAL DERIVATIVES
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Notation For Partial Derivative
The partial derivative with respect to y is denoted the same way as the partial derivative with respect to x:
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Example 1
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Example 2
Example 3
Solution We treat ƒ as a quotient. With y held constant, we get
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Example 4
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defines z as a function of the two independent variables x and y and the partial derivative exists.
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Second-Order Partial Derivatives
When we differentiate a function ƒ(x, y) twice, we produce its second-order derivatives. These derivatives are usuallydenoted by
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Example 5
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The Mixed Derivative Theorem
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Partial Derivatives of Still Higher Order
Example 6
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EXERCISES 14.3
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CHAIN RULEChain Rule for Functions of Two Independent Variables
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Example 1
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Chain Rule for Functions of Three Independent Variables
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Example 2
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Chain Rule for Two Independent Variables and ThreeIntermediate Variables
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Example 3
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Implicit Differentiation
Example 4: Find dy/dx if
If
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Total Differentiation• 1- If w = f(x,y,z), then the total differentiation is• dw = fx dx + fy dy + fz dz
• 2- If w = f(x,y,z), and x=x(t), y=y(t), z=z(t) then, the total differentiation is
•
dtdt
dz
z
wdt
dt
dy
y
wdt
dt
dx
x
wdw
3- If w = f(x,y,z), and x=x(r,s), y=y(r,s), z=z(r,s) then, the total differentiation is
dw = fx dx + fy dy + fz dzwhere
dss
xdr
r
xdx
ds
s
ydr
r
ydy
ds
s
zdr
r
zdz
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Exercises 14.4
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Extreme Values and Saddle Points
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First Derivative Test for Local Extreme Values
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Second Derivative Test for Local Extreme Values
Example 2
Example 3
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EXERCISES 14.7
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