Unformatted text preview: 248 CHAPTER 3. REALVALUED FUNCTIONS: DIFFERENTIATION
or
√
∂f
3
=√
∂ρ
22
√
3
∂f
=√
∂φ
2
1
∂f
=√ .
∂θ
2
The value and derivative of g(y ) at y = f 2, π , π =
43
1
g√
2
1
g′ √
2 1
√
2 are 1
= − ln 2;
2
√
=2 and from this we get
1
∂z
∂f
= g′ √
∂ρ
2 ∂ρ
√
√
3
√
=2
22
√
3
=
2
∂z
1
∂f
= g′ √
∂φ
2 ∂φ
√
√
3
=2√
2
√
=3
∂z
1
∂f
= g′ √
∂θ
2 ∂θ
√
1
=2√
2
= 1.
Note that this formula could have been found directly, using
Deﬁnition 3.3.3 (Exercise 7): the substantive part of the proof above was
to show that the composite function is diﬀerentiable. ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
 Fall '08
 ALL
 Calculus, Derivative

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