Chapter 2 Section 3. Partial Derivatives

Transcription

1 Chapter Section 3 Partial Derivatives

2 Deinition. Let be a unction o two variables and. The partial derivative o with respect to is the unction, denoted b D1 1 such that its value at an point (,) in the domain o is given b D 1 i this limit eists., lim h0,, h h

3 Deinition. Let be a unction o two variables and. The partial derivative o with respect to is the unction, denoted b D such that its value at an point (,) in the domain o is given b D i this limit eists., lim h0,, h h

4 Eercise., 3 5 Given that. Find and. D 1 1, lim h0, lim lim h0 h0, h,, 3 h h 3h lim 3 3 h h0 h

5 Eercise., 3 5 Given that. Find and. lim h0 1, D, lim h0,, h, 3 h h h 3h h h lim lim 3 h h0 h h0 h 3

6 REMARKS D, lim h0 D, lim D, lim 0 0 D, lim h0 0,, h h,, 0 0 0,, 0,, h h 0

7 P 1,,..., n Deinition. Let be a point in and be a unction o n variables. The partial derivative o with respect to k is the unction, denoted b such that its unction value at an point P in the domain o is given b D k P lim h0 i this limit eists. D k n R 1,,..., n,,...,,..., h P 1 k h n

8 Eample. Let, Find and 1,, Dierentiate with respect to ; Treating as a constant. 1, 6 4

9 Eample. Let Find, and 1,, Dierentiate with respect to ; Treating as a constant., 3 3 4

10 Chain Rule I u is a dierentiable unction o and deined b u,, where F r s G r s,,, and,,, all eist, then is a unction o and and u u u r r r r s r s u r s, u u u s s s

11 Let Find and. 3 Eercise. u 3 4, r, u u r s r s u r u r u r r 1 s

12 Let Find and. 3 Eercise. u 3 4, r, u u r s r s u s u s u s r s

13 Eercise. Let u z where sin cos sin sin u, Find and. u and u z cos u u u u z z

14 Let Find Eercise., e 3sin Arc tan and 1,, 1, e 3cos 1 1 1

15 Let Find Eercise., e 3sin Arc tan and 1,,, 3cos 1 1 1

16 Eercise. Let Find 3 3, ln and 1,, 1,

17 Eercise. Let Find 3 3, ln and 1,,, ln 1

18 Eercise. Let Find, log sec tan 3 and 1,, 1, 1 sec tan ln 3 sec tan tan 1 sec tan ln 3, sec sec

19 Geometric Interpretation. Let S be a surace given b. Consider a plane given b which intersects S at a curve C. z, 0 and Suppose is on C. P0 0, 0, z0 1, : slope o the tangent line to the curve C at P 0.

20 Let S be a surace given b. Consider a plane given b which intersects S at a curve C. Geometric Interpretation. z, 0 and Suppose is on C. P0 0, 0, z0, : slope o the tangent line to the curve C at P 0.

21 Let u ln e du dz Find. Total Derivative where csc z, cot z du dz u d dz u d dz ln e csc zcot z e csc z

22 Implicit Dierentiation Let and 3 tan 4 0 Find. d d

23 Implicit Dierentiation Theorem I is a dierentiable unction o the single variable such that = () and is deined implicitl b the equation F (, ) = 0, then i F is dierentiable and, then F, 0 d F, d F,

24 Higher Order Partial Derivatives

25 Second-Order Partial Derivative When we dierentiate a unction twice, we produce its second-order derivatives. These derivatives are usuall denoted b, The deining equations are

26 Second-Order Partial Derivative Let w w 3 w 4 3 w w w 6 3 1

27 Second-Order Partial Derivative Let w w 3 w w w w 6 3 1

28 Second-Order Partial Derivative, e 3sin Arc tan Let Find all o its second order partial derivatives., e 3 cos, e 3 sin, 3cos 3 sin

29 Second-Order Partial Derivative, e 3sin Arc tan Let, 3 cos, 3 sin , 3cos 3 sin

30 Second-Order Partial Derivative Theorem. The Mied Derivative Theorem I, and its partial derivatives Are deined throughout an open region containing a point ab,, then ab,,,, and are all continuous at,, a b a b

31 Second-Order Partial Derivative (¼ paper) Veri that given, ln ln

32 Remark. Higher-Order Partial Derivative There is no theoretical limit to how man times we can dierentiate a unction as long as the derivatives eist

33 Eercise. Let, cos Higher-Order Partial Derivative. Find, sin, 4 cos, 4 cos 3 8 sin

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