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Chapter 14: Problem 45

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

Short Answer

Expert verified
The partial derivatives are \( f_x(x, y) = y^2 - 3x^2y \) and \( f_y(x, y) = 2xy - x^3 \).

Step by step solution

01

Understand the Function

The given function is \( f(x, y) = xy^2 - x^3y \). We need to find the partial derivatives \( f_x(x, y) \) and \( f_y(x, y) \) using the definition of partial derivatives as limits.
02

Partial Derivative with respect to x

To find the partial derivative \( f_x(x, y) \), we consider \( y \) as a constant and differentiate with respect to \( x \):\[ f_x(x, y) = \lim_{h \to 0} \frac{f(x + h, y) - f(x, y)}{h} \]Substitute \( f(x, y) \) and simplify:\[ f(x+h, y) = (x+h)y^2 - (x+h)^3y \]\[ f(x + h, y) - f(x, y) = xy^2 + hy^2 - (x^3y + 3x^2hy + 3xh^2y + h^3y) - (xy^2 - x^3y) \]Simplify:\[ hy^2 - 3x^2hy - 3xh^2y - h^3y \]Divide by \( h \) and take the limit:\[ \lim_{h \to 0} \left( y^2 - 3x^2y - 3xhy - h^2y \right) = y^2 - 3x^2y \]
03

Partial Derivative with respect to y

To find the partial derivative \( f_y(x, y) \), we consider \( x \) as a constant and differentiate with respect to \( y \):\[ f_y(x, y) = \lim_{h \to 0} \frac{f(x, y + h) - f(x, y)}{h} \]Substitute \( f(x, y) \) and simplify:\[ f(x, y+h) = x(y+h)^2 - x^3(y+h) \]\[ f(x, y+h) - f(x, y) = x(y^2 + 2yh + h^2) - x^3(y + h) - (xy^2 - x^3y) \]Simplify:\[ 2xyh + xh^2 - hx^3 \]Divide by \( h \) and take the limit:\[ \lim_{h \to 0} \left( 2xy + xh - x^3 \right) = 2xy - x^3 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Limit Definition
The limit definition is a fundamental concept when it comes to finding derivatives. It's like zooming in on a curve until it looks like a straight line. For partial derivatives, we're doing this zooming for a multi-variable function. The key formula for finding a partial derivative with respect to a variable, say \( x \), involves taking the limit as \( h \) approaches 0 of \( \frac{f(x+h, y) - f(x, y)}{h} \).
  • Here, \( x \) is the variable of interest, while \( y \) is held constant.
  • The idea is to observe how the function changes as we make small adjustments to \( x \), while \( y \) stays the same.
  • Similarly, to find a partial derivative with respect to \( y \), you switch roles and consider \( x \) as a constant.
Understanding and applying this limit is crucial as it forms the basis for calculating partial derivatives. This process reveals the function's sensitivity to changes in one variable while keeping others fixed.
Functions of Several Variables
Functions of several variables, like \( f(x, y) = xy^2 - x^3y \), are an extension of single-variable functions to more dimensions. Instead of just being reliant on one variable, these functions depend on multiple inputs. This opens up a new world of possibilities and complexities, which makes them very intriguing.
  • Each variable in the function can affect the output in different ways.
  • This causes the surface the function represents to be a lot more than just a line or a simple curve.
  • Visualizing these can be like thinking of a landscape where each point has its own height (value of the function).
When working with such functions, it's essential to grasp how varying one variable influences others. Partial derivatives come into play here, allowing us to explore these relationships by isolating individual effects.
Differentiation
Differentiation is the process of calculating a derivative, which in the context of multi-variable functions, involves taking partial derivatives. It measures how a function changes as its input variables are slightly varied.
  • The partial derivative \( f_x(x, y) \) tells us the rate of change of \( f \) with respect to \( x \), while keeping \( y \) constant.
  • Similarly, \( f_y(x, y) \) gives the rate of change with respect to \( y \), with \( x \) held constant.
  • This method can be used to find tangent planes and optimize functions of several variables.
Differentiation with partial derivatives is powerful because it lets us analyze the effect of each variable separately. This can help us understand complex systems modeled by functions of several variables, making it a key tool in mathematical analysis and applications.

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Most popular questions from this chapter

The paraboloid \(z=6-x-x^{2}-2 y^{2}\) intersects the plane \(x=1\) in a parabola. Find parametric equations for the tangent line to this parabola at the point \((1,2,-4) .\) Use a computer to graph the paraboloid, the parabola, and the tangent line on the same screen.

If \(f(x, y)=\sqrt[3]{x^{3}+y^{3}},\) find \(f_{x}(0,0)\)

(a) If your computer algebra system plots implicitly defined curves, use it to estimate the minimum and maximum values of \(f(x, y)=x^{3}+y^{3}+3 x y\) subject to the constraint \((x-3)^{2}+(y-3)^{2}=9\) by graphical methods. (b) Solve the problem in part (a) with the aid of Lagrange multipliers. Use your CAS to solve the equations numerically. Compare your answers with those in part (a).

Find the limit, if it exists, or show that the limit does not exist. $$ \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x^{2} y^{2} z^{2}}{x^{2}+y^{2}+z^{2}} $$

Find the maximum and minimum values of \(f\) subject to the given constraints. Use a computer algebra system to solve the system of equations that arises in using Lagrange multipliers. (If your CAS finds only one solution, you may need to use additional commands.) $$ f(x, y, z)=y e^{x-z} ; \quad 9 x^{2}+4 y^{2}+36 z^{2}=36, x y+y z=1 $$
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