Problem 43 - Calculus Early Transcendentals

Question

Use the definition of partial derivatives as limits \((4)\) to find \(f_{x}(x, y)\) and \(f_{y}(x, y) .\) $$f(x, y)=x y^{2}-x^{3} y$$

Step-by-Step Solution

Answer

\( f_{x}(x, y) = y^{2} - 3x^{2}y \) and \( f_{y}(x, y) = 2xy - x^{3} \).

Step 1: Write the formula for the partial derivative of f with respect to x

The partial derivative of \( f \) with respect to \( x \), \( f_{x}(x, y) \), is found by treating \( y \) as a constant and differentiating \( f \) with respect to \( x \). We will use the limit definition: \[f_{x}(x, y) = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}\]

Step 2: Evaluate f(x+h, y)

Substitute \( x+h \) into the function for \( x \), keeping \( y \) constant: \[f(x+h, y) = (x+h)y^{2} - (x+h)^{3}y\]Simplify:\[x y^{2} + hy^{2} - (x^3 + 3x^2h + 3xh^2 + h^3)y\]

Step 3: Simplify and find the limit for f_{x}(x, y)

Subtract \( f(x, y) \) from \( f(x+h, y) \) and simplify:\[f(x+h, y) - f(x, y) = hy^2 - (3x^2h + 3xh^2 + h^3)y\]Divide by \( h \): \[\frac{h(y^2 - 3x^2y) - 3xh^2y - h^3y}{h} = y^2 - 3x^2y - 3xhy - h^2y\]Take the limit as \( h \to 0 \): \[f_{x}(x, y) = y^2 - 3x^2y\]

Step 4: Write the formula for the partial derivative of f with respect to y

The partial derivative of \( f \) with respect to \( y \), \( f_{y}(x, y) \), is found by treating \( x \) as a constant and differentiating \( f \) with respect to \( y \). Use the limit definition:\[f_{y}(x, y) = \lim_{k \to 0} \frac{f(x, y+k) - f(x, y)}{k}\]

Step 5: Evaluate f(x, y+k)

Substitute \( y+k \) into the function for \( y \), keeping \( x \) constant:\[f(x, y+k) = x(y+k)^2 - x^3(y+k)\]Expand:\[xy^2 + 2xyk + xk^2 - x^3y - x^3k\]

Step 6: Simplify and find the limit for f_{y}(x, y)

Subtract \( f(x, y) \) from \( f(x, y+k) \) and simplify:\[f(x, y+k) - f(x, y) = 2xyk + xk^2 - x^3k\]Divide by \( k \): \[\frac{k(2xy) + xk^2 - x^3k}{k} = 2xy + xk - x^3\]Take the limit as \( k \to 0 \): \[f_{y}(x, y) = 2xy - x^3\]

Step 7: Conclusion: Summary of Partial Derivatives

The partial derivatives are: \[f_{x}(x, y) = y^{2} - 3x^{2}y\]\[f_{y}(x, y) = 2xy - x^{3}\]These represent the rates of change of the function with respect to \( x \) and \( y \).

Key Concepts

Limit Definition

In calculus, the limit definition is an essential tool for understanding how functions change. When applied to partial derivatives, it helps determine how a function behaves when one of its input values is slightly changed, keeping other inputs constant. The standard form of deriving a partial derivative using the limit definition is:
\[ f_x(x, y) = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}\]This takes the form of a derivative along one axis, where contributions from changes along other axes are ignored.

• The limit examines how the function changes as the increment \( h \) approaches zero.
• The difference quotient \( \frac{f(x+h, y) - f(x, y)}{h} \) measures the average rate of change of the function between \( x \) and \( x+h \).

As such, this concept is crucial when exploring partial derivatives, since it lays the foundation for calculations that involve more complex changes in multi-variable functions.

Differentiation with Respect to x

The process of finding a partial derivative with respect to \( x \) involves differentiating the function as if \( y \) were a fixed constant. This means we treat the function as a regular single-variable function with \( x \) being the variable of interest.
In the given function, \( f(x, y) = xy^2 - x^3y \), we assess the change concerning \( x \).

• Substitute \( x+h \) into the function, giving us \( (x+h)y^2 - (x+h)^3y \).
• Simplify to get \( xy^2 + hy^2 - (x^3 + 3x^2h + 3xh^2 + h^3)y \).
• After applying the difference quotient and taking \( h \to 0 \), the result is \( y^2 - 3x^2y \).

This partial derivative quantifies how \( f(x, y) \) changes with variation in \( x \) while keeping \( y \) constant.

Differentiation with Respect to y

Like differentiating with respect to \( x \), when differentiating with respect to \( y \), \( x \) is considered constant. This highlights how the function varies as only \( y \) changes.
Given the function \( f(x, y) = xy^2 - x^3y \), we observe how it changes as \( y \) changes alone.

• Replace \( y\) with \( y+k \) leading to \( x(y+k)^2 - x^3(y+k) \).
• Expanding gives \( xy^2 + 2xyk + xk^2 - x^3y - x^3k \).
• After simplifying the quotient and letting \( k \to 0 \), we arrive at \( 2xy - x^3 \).

This derivative indicates the rate of change of \( f(x, y) \) for incremental changes in \( y \), while \( x \) remains fixed.

Rates of Change

Partial derivatives provide insights into rates of change for multivariable functions. By calculating \( f_x(x, y) \) and \( f_y(x, y) \), we understand how \( f(x, y) \) reacts to small adjustments in \( x \) or \( y \). This can be particularly enlightening:

• In the result, \( f_x(x, y) = y^2 - 3x^2y \) highlights the function's sensitivity to \( x \) increments amid constant \( y \).
• Similarly, \( f_y(x, y) = 2xy - x^3 \) points out \( f(x, y) \)'s response when adjusting \( y \) while keeping \( x \) unvaried.

Grasping these rates of change is vital for fields like physics, engineering, and economics, where understanding variables’ interdependencies is crucial. Such derivatives help in constructing models that predict how systems will behave under varying conditions.