Problem 112 - Calculus Volume 3

Question

For the following exercises, calculate the partial derivative using the limit definitions only. $$ \frac{\partial z}{\partial x}\( for \)z=x^{2}-3 x y+y^{2}$$

Step-by-Step Solution

Answer

The partial derivative \( \frac{\partial z}{\partial x} \) is \( 2x - 3y \).

Step 1: Understand the Problem

We are asked to find the partial derivative of the function \( z = x^2 - 3xy + y^2 \) with respect to \( x \) using the limit definition. The partial derivative with respect to \( x \) treats \( y \) as a constant.

Step 2: Recall the Limit Definition of Partial Derivative

The partial derivative of \( z \) with respect to \( x \) is given by the limit:\[ \frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h} \]where \( f(x, y) = x^2 - 3xy + y^2 \).

Step 3: Substitute into the Definition

Substitute \( f(x + h, y) = (x + h)^2 - 3(x + h)y + y^2 \) and \( f(x, y) = x^2 - 3xy + y^2 \) into the limit expression. The difference becomes:\[ \frac{(x + h)^2 - 3(x + h)y + y^2 - (x^2 - 3xy + y^2)}{h} \].

Step 4: Simplify the Expression

Expand \( (x + h)^2 \) to get \( x^2 + 2xh + h^2 \) and \( -3(x + h)y \) to \( -3xy - 3hy \). Substitute these back into the expression:\[ \frac{x^2 + 2xh + h^2 - 3xy - 3hy + y^2 - x^2 + 3xy - y^2}{h} \] simplifies to \[ \frac{2xh + h^2 - 3hy}{h} \].

Step 5: Cancel Terms and Take the Limit

Factor out \( h \) in the numerator: \[ \frac{h(2x + h - 3y)}{h} \]. Cancel \( h \) from the numerator and denominator, yielding \( 2x + h - 3y \). Now, take the limit as \( h \to 0 \):\[ \lim_{h \to 0} (2x + h - 3y) = 2x - 3y \].

Step 6: Final Result

The partial derivative of \( z \) with respect to \( x \) using the limit definition is \( \frac{\partial z}{\partial x} = 2x - 3y \).

Key Concepts

Limit Definition of Partial Derivatives

The concept of partial derivatives involves taking the derivative of a function with multiple variables in terms of just one variable while holding the others constant.
A fundamental way to approach this is using the limit definition.
Just as with regular derivatives, partial derivatives can be calculated using limits to find the slope of the tangent plane at a point on the surface described by the function.
To calculate the partial derivative of a function \( f(x, y) = x^2 - 3xy + y^2 \) with respect to \( x \), we use:

• \( \frac{\partial z}{\partial x} = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h} \)
• This formula finds the derivative of \( f(x, y) \) as if \( y \) is a constant number, focusing only on changes in \( x \).

This approach effectively breaks down complex multi-variable problems into simpler, single-variable ones by treating other variables as constants.

Differentiation in Calculus

Differentiation is a core concept in calculus that involves finding the rate at which a function changes at any point.
For functions of two variables like \( z = x^2 - 3xy + y^2 \), we perform differentiation partially. We differentiate with respect to one variable, while treating the others as constants.
In the context of this exercise:

• By finding \( \frac{\partial z}{\partial x} \), we only look at how changes in \( x \) impact \( z \), while \( y \) remains unchanged.
• Such differentiation provides insights into the geometry of a function. It shows the slope of the surface along the \( x \) direction.

Differentiation, therefore, allows us to explore how sensitive a given quantity is to changes in one particular direction.

Understanding Multivariable Calculus

Multivariable calculus extends the principles of ordinary calculus to functions of several variables.
It involves studying functions where inputs have more than one dimension.
In the equation \( z = x^2 - 3xy + y^2 \), both \( x \) and \( y \) are independent variables:

• For such problems, partial derivatives allow us to examine how each individual variable affects the function.
• This greatly aids in appreciating the complex interplay between variables in many real-life scenarios where multiple factors influence outcomes.

Understanding and mastering multivariable calculus opens the door to advanced concepts and applications such as optimization, curve sketching in higher dimensions, and modeling physical phenomena.

Approaches to Calculus Problem Solving

Effective calculus problem solving involves breaking down the problem into manageable steps. Particularly for multivariable functions, this means focusing on one variable at a time using partial derivatives.
Here's a step-by-step strategy:

• Begin by clearly identifying the function and what is being asked, such as which variable's partial derivative to find.
• Substitute into the limit definition, carefully setting up differences between function values with incremental changes.
• Expand, simplify, and cancel terms as necessary to reveal a clear path to the solution.
• Finally, systematically take the limit to find the precise rate of change.

By following these steps, students can break complex problems into straightforward calculations, reinforcing their understanding of calculus principles.