Deriving the Total Derivative Formula for Multivariable Functions

In mathematics, particularly in the field of calculus, the concept of a total derivative is a fundamental tool used to understand how a multivariable function changes as its variables change. This article will delve into the process of deriving the total derivative of a function (u(x, y)) with respect to a parameter (t), which is often used to model scenarios where both (x) and (y) vary over time.

Introduction to Total Derivative

Consider a function (u(x, y)), where both (x) and (y) are functions of another variable (t). In most practical applications, (x(t)) and (y(t)) represent the positions of objects or quantities that change with time. The key idea is to find how the function (u(x(t), y(t))) changes as time (t) progresses.

Step-by-Step Derivation

Let's break down the process of deriving the total derivative formula in a step-by-step manner.

Step 1: Incremental Change in the Function

The total differential of (u(x, y)) is given by:

(displaystyle mathbf{mathrm{d}u frac{partial u}{partial x} mathrm{d}x frac{partial u}{partial y} mathrm{d}y})

Here, (frac{partial u}{partial x}) and (frac{partial u}{partial y}) represent the partial derivatives of (u) with respect to (x) and (y) respectively, and (mathrm{d}x) and (mathrm{d}y) denote the small changes in (x) and (y).

Step 2: Expressing (mathrm{d}x) and (mathrm{d}y) in Terms of (t)

Since both (x) and (y) are functions of (t), we can express (mathrm{d}x) and (mathrm{d}y) as:

(displaystyle mathrm{d}x frac{mathrm{d}x}{mathrm{d}t} mathrm{d}t)

(displaystyle mathrm{d}y frac{mathrm{d}y}{mathrm{d}t} mathrm{d}t)

Step 3: Substituting and Simplifying

Substitute these expressions into the total differential formula:

(displaystyle mathrm{d}u frac{partial u}{partial x} left(frac{mathrm{d}x}{mathrm{d}t} mathrm{d}t right) frac{partial u}{partial y} left(frac{mathrm{d}y}{mathrm{d}t} mathrm{d}t right))

Simplifying further, we get:

(displaystyle mathrm{d}u left(frac{partial u}{partial x} frac{mathrm{d}x}{mathrm{d}t} frac{partial u}{partial y} frac{mathrm{d}y}{mathrm{d}t} right) mathrm{d}t)

To find the total derivative of (u) with respect to (t), we divide both sides by (mathrm{d}t):

(displaystyle frac{mathrm{d}u}{mathrm{d}t} frac{partial u}{partial x} frac{mathrm{d}x}{mathrm{d}t} frac{partial u}{partial y} frac{mathrm{d}y}{mathrm{d}t})

This final expression represents the total derivative of (u(x(t), y(t))) with respect to (t).

Conclusion

Understanding the total derivative is crucial in many areas of science and engineering, including physics, economics, and engineering dynamics. By breaking down the process into manageable steps and using clear notational representations, we can derive the total derivative of a multivariable function with respect to a parameter, such as time.

The process involves using partial differentiation to account for changes in the function due to changes in each of its variables, and then combining these changes using the chain rule. This technique provides a powerful tool for analyzing complex systems where multiple variables are interrelated.

Keywords: total derivative, multivariable function, partial derivatives, chain rule