Calculating Partial Derivatives of a Multivariate Function at a Point

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SUMMARY

The discussion centers on calculating the partial derivatives of the multivariate function f(x,y,z) = (z³/y, x³/z) at the point (1,2,3). Participants confirm that the correct approach involves finding the Jacobian matrix, which represents the partial derivatives of the function. The solution requires constructing a 2x3 matrix of these derivatives and evaluating it at the specified point. The consensus is that this interpretation aligns with standard practices in multivariable calculus.

PREREQUISITES
  • Understanding of multivariable calculus concepts, specifically partial derivatives.
  • Familiarity with the Jacobian matrix and its applications.
  • Basic knowledge of matrix operations and evaluation.
  • Proficiency in evaluating functions at specific points.
NEXT STEPS
  • Study the properties and applications of the Jacobian matrix in multivariable calculus.
  • Learn how to compute partial derivatives for more complex functions.
  • Explore the relationship between Jacobian matrices and transformations in higher dimensions.
  • Practice problems involving the evaluation of multivariate functions at given points.
USEFUL FOR

Students studying multivariable calculus, educators teaching calculus concepts, and anyone seeking to deepen their understanding of partial derivatives and Jacobian matrices.

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Homework Statement



Calculate the derivative of f(x,y,z)=([tex]\frac{z^3}{y}[/tex] , [tex]\frac{x^3}{z}[/tex]) at (1,2,3)

Homework Equations

The Attempt at a Solution



Okay guys and gals, this problem was on my final today. It was the only problem I had a gutsy but unsure feeling about. The actual answer itself does not matter to me, rather the concept of understanding is.

I basically interpreted the problem this way:

Find the partial derivatives with respect to x,y,z for the two functions of f. So it is a 2 x 3 matrices. Plug in the points and you are done.

Did anyone else get this interpretation of the problem/solution?
 
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