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Partial Derivatives of Surface Equations and Clairaut's Theorem

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Find all the second partial derivatives.

47. f(x, y) = x^4 - 3(x^2)(y^3)

Verify that the conclusion of Clairaut's Theorem holds, that is, u_xy = u_yx.

55. u = ln[sqrt(x^2 + y^2)]

Find the indicated partial derivative.

59. f(x, y, z) = cos(4x + 3y + 2z); f_xyz, f_yzz

89. If f(x,y) = x(x^2 + y^2)^-3/2 * e^(sin(x^2 * y)), find (f_x)(1, 0). Hint: Instead of finding (f_x)(x,y) first, not that it's easier to use Equation 1 or Equation 2.

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