Is the Chain Rule Application for Second Partial Derivatives Correct?

Thread starter STEMucator
Start date Oct 13, 2012
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SUMMARY

The discussion centers on the application of the Chain Rule for computing second partial derivatives in the context of a function w(s,t) defined as w(s,t) = f(x,y), where x = p(s,t) and y = g(s,t). The user successfully computes the first partial derivative w_s using the formula w_s = f_x x_s + f_y y_s. However, uncertainty arises in the computation of the second mixed partial derivative w_{st}, which is expressed as w_{st} = (f_{xx} x_t + f_{xy} y_t)x_s + f_x x_{st} + (f_{xy} x_t + f_{yy} y_t)y_s + f_y y_{st}. The user seeks verification of this expression to ensure accuracy before an upcoming exam.

PREREQUISITES

Understanding of the Chain Rule in multivariable calculus
Familiarity with partial derivatives and their notation
Knowledge of functions of multiple variables
Experience with mixed partial derivatives

NEXT STEPS

Review the Chain Rule for multiple variables in detail
Practice computing mixed partial derivatives with various functions
Study examples of second partial derivatives in multivariable calculus
Explore the implications of Clairaut's theorem on mixed partial derivatives

USEFUL FOR

Students studying multivariable calculus, particularly those preparing for exams involving partial derivatives and the Chain Rule. This discussion is also beneficial for educators looking to clarify concepts related to second partial derivatives.

Oct 13, 2012

STEMucator

Homework Helper

Homework Statement

I'm curious to know if I'm actually doing this correctly.

Suppose f(x,y) is a function where x = p(s,t) and y = g(s,t) so that w(s,t) = f(x,y).

Compute w_s and then w_st

Homework Equations

Chain Rule.

The Attempt at a Solution

So! Let's compute w_s first. Whenever I use a subscript I refer to the partial with respect to that variable.

[itex]w_s = f_x x_s + f_y y_s[/itex]

That was nice and easy... now for the hard part that I'm not sure of. Let's compute w_st :

[itex]w_{st} = (f_{xx} x_t + f_{xy} y_t)x_s + f_x x_{st} + (f_{xy} x_t + f_{yy} y_t)y_s + f_y y_{st}[/itex]

I think this is correct, but with all the variables floating around I'm not entirely sure I didn't miss anything. I would appreciate it very much if someone could verify this for me as I don't want to have a 'blah' moment when my exam happens.