Question

Find \(\partial y / \partial x_{1}\) and \(\partial y / \partial x_{2}\) for each of the following functions: \((a) y=2 x_{1}^{3}-11 x_{1}^{2} x_{2}+3 x_{2}^{2}\) (b) \(y=7 x_{1}+6 x_{1} x_{2}^{2}-9 x_{2}^{3}\) (c) \(y=\left(2 x_{1}+3\right)\left(x_{2}-2\right)\) \((d) y=\left(5 x_{1}+3\right) /\left(x_{2}-2\right)\)

Short answer

Each partial derivative was found by treating the other variable as a constant, applying the rules of basic calculus and algebra.

Step-by-step solution

Differentiate Part (a) with respect to x1

For the function \( y = 2x_1^3 - 11x_1^2x_2 + 3x_2^2 \), find \( \frac{\partial y}{\partial x_1} \) by treating \( x_2 \) as a constant. \( \frac{\partial y}{\partial x_1} = 6x_1^2 - 22x_1x_2 \)

Differentiate Part (a) with respect to x2

For the same function, find \( \frac{\partial y}{\partial x_2} \) by treating \( x_1 \) as a constant. \( \frac{\partial y}{\partial x_2} = -11x_1^2 + 6x_2 \)

Differentiate Part (b) with respect to x1

For the function \( y = 7x_1 + 6x_1x_2^2 - 9x_2^3 \), find \( \frac{\partial y}{\partial x_1} \) by treating \( x_2 \) as a constant. \( \frac{\partial y}{\partial x_1} = 7 + 6x_2^2 \)

Differentiate Part (b) with respect to x2

For the same function, find \( \frac{\partial y}{\partial x_2} \) by treating \( x_1 \) as a constant. \( \frac{\partial y}{\partial x_2} = 12x_1x_2 - 27x_2^2 \)

Differentiate Part (c) with respect to x1

For the function \( y = (2x_1 + 3)(x_2 - 2) \), use the product rule to find \( \frac{\partial y}{\partial x_1} \).\( \frac{\partial y}{\partial x_1} = (x_2 - 2) \cdot 2 = 2(x_2 - 2) = 2x_2 - 4 \)

Differentiate Part (c) with respect to x2

Use the product rule for the same function to find \( \frac{\partial y}{\partial x_2} \).\( \frac{\partial y}{\partial x_2} = (2x_1 + 3) \cdot 1 = 2x_1 + 3 \)

Differentiate Part (d) with respect to x1

For the function \( y = \frac{5x_1 + 3}{x_2 - 2} \), treat \( x_2 \) as a constant to get the derivative. \( \frac{\partial y}{\partial x_1} = \frac{5}{x_2 - 2} \)

Differentiate Part (d) with respect to x2

For the same function, use the quotient rule to find \( \frac{\partial y}{\partial x_2} \).\( \frac{\partial y}{\partial x_2} = \frac{-(5x_1 + 3)}{(x_2 - 2)^2} \)

Key concepts

Multivariable Calculus

Multivariable calculus extends the concepts of calculus to functions with more than one variable. Instead of focusing on functions with a single input, multivariable calculus deals with functions that have multiple inputs and yields insights into how these inputs interact and affect the output. This branch of calculus is essential in understanding the nature and behavior of complex systems in fields like physics, engineering, and economics.

• Partial Derivatives: These are derivatives of functions with several variables, taken with respect to one variable while treating other variables as constants. They give us information about how the function changes as each variable changes individually.
• Applications: Multivariable calculus is used for optimizing functions subject to constraints, analyzing dynamic systems, and modeling real-world phenomena. It's crucial for design, data analysis, and computational simulations.

For students, multivariable calculus can seem daunting due to increased complexity, but breaking down the problem into parts can make tackling it much more manageable.

Function Differentiation

Differentiating a function is the process of finding the derivative, which measures how a function changes as its input changes. In the context of multivariable calculus, function differentiation takes on a broader meaning. Here, we often deal with partial derivatives, which allow us to understand multidirectional rates of change.

• Understanding Rates of Change: In single-variable calculus, a derivative tells us the rate at which a quantity changes. In multivariable contexts, partial derivatives tell us how changing one variable while keeping others fixed affects the overall function.
• Applications in Real Life: This becomes useful in scenarios like physics when analyzing how changing one aspect of a system (like temperature) affects other properties (like pressure).

Trust your foundational knowledge of single-variable differentiation, and apply it contextually, keeping other variables fixed. This approach tends to simplify complex functions.

Quotient Rule

The quotient rule is a technique for finding derivatives of functions that are expressed as a quotient of two simpler functions.
When encountering a function like \[ \frac{u(x)}{v(x)} \], we use the quotient rule for differentiation:
\[\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v \cdot u' - u \cdot v'}{v^2}\]In a multivariable setting, to find a partial derivative, we treat the function as a quotient of expressions depending on several variables.

• Simultaneous Derivatives: In partial derivatives, even when applying the quotient rule, remember that you're focusing on one variable at a time while treating others as constants.
• Real-World Application: For instance, if how two factors divide affects output (like cost per unit), you'd use the quotient rule to find how changing one component affects this ratio.

The quotient rule might be trickier than the product rule due to its formula, but careful application can lead to simplifying complex expressions, giving clear insights into the rate of change.

Product Rule

The product rule is a vital tool used to find the derivative of a product of two functions. In multivariable calculus, the product rule can be applied to partial derivatives to understand their behavior.
For two functions, say\( u \) and\( v \), the derivative of their product concerning some variable is given by:\[\frac{d}{dx}(uv) = u'v + uv'\]In the context of multivariable functions, each partial derivative can be broken into smaller, more manageable parts using the product rule, making it similar to Lego pieces fitting together.

• Reducing Complexity: These rules are particularly useful in multivariable calculus where functions can have complex taxa interaction like polynomial models or layered systems.
• Handling Each Variable: Treat every variable independently when differentiating a function. This approach ensures better manageability and clarity when dealing with intricate equations.

Grasping the concept of the product rule in partial differentiation opens up new avenues for mathematical modeling and analysis in multi-dimensional spaces.