Understanding the Derivation of the Multivariable Chain Rule Using Limits

The multivariable chain rule is a crucial tool in calculus, particularly in scenarios where functions of several variables depend on a common variable. This article aims to provide a clear and detailed explanation of how the multivariable chain rule can be derived by combining the definition of a derivative and limits.

Step-by-Step Derivation of the Multivariable Chain Rule

To derive the multivariable chain rule for a function ftxt yt with respect to t, we will use the fundamental concepts of limits and the definition of a derivative.

Step 1: Definition of the Derivative

The derivative of a function gt at a point t_0 is defined as:

gt_0 lim_{h to 0} (gt_0 h - gt_0) / h

In this context, we aim to find the derivative of ftxt yt with respect to t.

Step 2: Apply the Definition to ftxt yt

Let gt fxt yt. The derivative of gt at t_0 is given by:

gt_0 lim_{h to 0} (fxt_0 h yt_0 h - fxt_0 yt_0) / h

Step 3: Use the Increment of h

We can express xt_0 h and yt_0 h using the continuity of f and the differentiability of xt and yt around t_0. This gives:

xt_0 h xt_0 xt_0h oh, yt_0 h yt_0 yt_0h oh

where oh represents terms that go to zero faster than h as h to 0.

Step 4: Use Taylor Expansion for f

We can expand f using a Taylor series around the point xt_0 yt_0:

fxt_0 h yt_0 h fxt_0 yt_0 f_x_{xt_0 yt_0} (h - xt_0) f_y_{xt_0 yt_0} (h - yt_0) osime sqrt{h - xt_0^2 h - yt_0^2}

where osime is a term that approaches zero as h approaches zero.

Step 5: Substitute the Increments

Substituting the increments for xt_0 h and yt_0 h:

fxt_0 h yt_0 h fxt_0 yt_0 f_x_{xt_0 yt_0} (h - xt_0) f_y_{xt_0 yt_0} (h - yt_0) oh

Step 6: Combine Terms

Combining the terms, focusing on the leading ones:

fxt_0 h yt_0 h fxt_0 yt_0 f_x_{xt_0 yt_0} (h - xt_0) f_y_{xt_0 yt_0} (h - yt_0) oh

Step 7: Substitute Back into the Derivative Definition

Substituting back into the limit for gt_0:

gt_0 lim_{h to 0} (fxt_0 yt_0 f_x_{xt_0 yt_0} (h - xt_0) f_y_{xt_0 yt_0} (h - yt_0) oh - fxt_0 yt_0) / h

This simplifies to:

gt_0 lim_{h to 0} (f_x_{xt_0 yt_0} (h - xt_0) f_y_{xt_0 yt_0} (h - yt_0) oh) / h

Step 8: Evaluate the Limit

As h to 0, the oh/h term goes to zero, and we have:

gt_0 f_x_{xt_0 yt_0} (h - xt_0) f_y_{xt_0 yt_0} (h - yt_0)

Thus, the chain rule for the function ftxt yt is given by:

????d/dt fxt yt f_x_{xt yt} d/dt x f_y_{xt yt} d/dt y

Conclusion

This is the multivariable chain rule for the case of ftxt yt with respect to t. Understanding the derivation using the definition of a derivative and limits is essential for grasping this powerful tool in calculus.