Labware - MA35 Multivariable Calculus - Three Variable Calculus

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Second Partial Derivative

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A second partial derivative is a partial derivative of a function which is itself a partial derivative of another function.

There are nine types of second partial derivatives for functions of three variables.

1. fxx(x,y,z) = Partial derivative of fx(x,y,z) with respect to x.

2. fyx(x,y,z) = Partial derivative of fx(x,y,z) with respect to y.

3. fzx(x,y,z) = Partial derivative of fx(x,y,z) with respect to z.

4. fxy(x,y,z) = Partial derivative of fy(x,y,z) with respect to x.

5. fyy(x,y,z) = Partial derivative of fy(x,y,z) with respect to y.

6. fzy(x,y,z) = Partial derivative of fy(x,y,z) with respect to z.

7. fxz(x,y,z) = Partial derivative of fz(x,y,z) with respect to x.

8. fyz(x,y,z) = Partial derivative of fz(x,y,z) with respect to y.

9. fzz(x,y,z) = Partial derivative of fz(x,y,z) with respect to z.

Third, fourth, fifth, and in general, nth partial derivatives for any positive integer n, exist as well.

Demos

Exercises

  • 1. Find the following second partial derivatives as functions of x, y, and z:
    • fxz(x, y, z), where f(x, y, z) = x2 + xy + xz
    • fyy(x, y, z), where f(x, y, z) = x2yz + xy2z + xyz2
    • fzy(x, y, z), where f(x, y, z) = zey
    • fxy(x, y, z), where f(x, y, z) = 1/(xyz)
  • 2. Suppose the demo above did not provide graphs of the first and second partial derivatives. How could you use the graph of f(x, y, z) to determine what the graph of fxy(x, y, z) would look like?
  • 3. Consider the function f(x, y, z) = Axy + Bxz + Cyz. Describe the relationships between the mixed second partial derivatives. Will these relationships exist for any function of three variables f(x, y, z) or are there exceptions?