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Find all the second-order partial derivatives of the following function. w = 6x sin (x y)\fThe three-dimensional Laplace equation 2 =0 is satisfied by steady-state temperature distributions dx day dz T= f(x,y,z) in space, by gravitational potentials, and by electrostatic potentials. Show that the function satisfies the three-dimensional Laplace equation. f ( x, y.z) = (x2 +y? +=2) -1/4 Find the second-order partial derivatives of f(x,y,z) with respect to x, y, and z, respectively. (3x2 - 2y? - 22 2) (x2 + y2 +23 ) (3y? - 2x- - 22 2) (x 7 + y?+2? ) 1 (32 2 - 2x2 - 2y?) A dz ( x 7 + +z-) (Simplify your answers. Use positive exponents only.)aw d'w In the one-dimensional wave equation, = C- , wave height is represented by w, x is the distance variable, t is the time variable, and c is the velocity with which the waves are propagated. Show that the function w = In (2x + 2ct) is a solution to the wave equation. 2x 2 To verify that for w = In (2x + 2ct), compute and and compare the results. at ax aw What definitions, properties, and rules are needed to find and - -? Select all that apply. ox A. The Definition of Differentiability LY B. The Chain Rule C. The Mixed Derivative Theorem D. The Product Rule dw C at x + ct ow dx X+ ct a w =0\fdw Use the functions below to (a) express - dt as a function of t, both by using the Chain Rule and by expressing w in dw terms of t and differentiating directly with respect to t. Then (b) evaluate dt at the given value of t. W =z - sin xy, x = 3t, y = In (t+7), z=el*3, t= -6 dw (a) Express dt as a function of t. dw 3t cos (3t Int + 7) - 3 In (t+ 7) cos| 3t In (t + 7)- 1+6 = +e dt t+7Consider the functions 2 = 5 ex In 1;. x: In [u cos v}: and y = u sin v. dz dz [a] Express E and E as functions of u and v both by using the Chain Rule and by expressing 2 directly in terms of u and v before differentiating. 32 :37. 11: [[1] Evaluate E and E at (1450 I? . dz [all Find each partial derivative needed to use the Chain Rule to nd 5. dz x dx 1 at: 5.-;- lug du u dz se" i _ _ = d" 5ll'l'lu' \"l!" 3' Express 2 directly in terms of u and v. FE az Using either method, du (Type an expression using u and v as the variables.) az Find each partial derivative needed to use the Chain Rule to find av az Ox E ax av az dy = = dy av az Using either method. av (Type an expression using u and v as the variables.)