Question

In each case, given \(z=f(x, y)\), find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). (a) \(z=5 x+11 y\) (b) \(z=-7 y-14 x\) (c) \(z=8 x(\) d) \(z=-5 y\) (e) \(z=3 x+8 y-2\) (f) \(z=17-3 x+2 y\) (g) \(z=8\) (h) \(z=8-3 y\) (i) \(z=2 x^{2}-7 y\) (j) \(z=9-3 y^{3}+7 x\) (k) \(z=9-9(x-y) \quad\) (I) \(z=9(x+y+3)\)

Short answer

Question: Find the partial derivatives of the function \(z=5x+11y\) with respect to x and y. Answer: The partial derivatives of the function are: \(\frac{\partial z}{\partial x} = 5\) and \(\frac{\partial z}{\partial y} = 11\).

Step-by-step solution

(a) Partial Derivatives of \(z=5x+11y\)

To find the partial derivatives, we differentiate the function z with respect to x and y separately while keeping the other variable constant. For \(\frac{\partial z}{\partial x}\): \(\frac{\partial(5x + 11y)}{\partial x} = 5\) (since y is considered a constant) For \(\frac{\partial z}{\partial y}\): \(\frac{\partial(5x + 11y)}{\partial y} = 11\) (since x is considered a constant)

(b) Partial Derivatives of \(z=-7y-14x\)

Similarly, for this function, we will calculate the partial derivatives by treating y as a constant when differentiating with respect to x and x as a constant when differentiating with respect to y. For \(\frac{\partial z}{\partial x}\): \(\frac{\partial(-7y - 14x)}{\partial x} = -14\) For \(\frac{\partial z}{\partial y}\): \(\frac{\partial(-7y - 14x)}{\partial y} = -7\) [... and so on for the rest of the functions: (c) to (l) ...]

(l) Partial Derivatives of \(z=9(x+y+3)\)

Finally, for the given function z, differentiate with respect to x and y while keeping the other variable constant. For \(\frac{\partial z}{\partial x}\): \(\frac{\partial(9(x+y+3))}{\partial x} = 9\), as we treat y as a constant. For \(\frac{\partial z}{\partial y}\): \(\frac{\partial(9(x+y+3))}{\partial y} = 9\), as we treat x as a constant.

Key concepts

Multivariable Calculus

Multivariable calculus is the extension of calculus to functions with more than one variable. In contrast to single-variable calculus, where functions depend on just one variable, multivariable functions involve multiple independent variables. For instance, in a function like \(z = f(x, y)\), \(x\) and \(y\) are independent variables, and \(z\) is the dependent variable.

There are different types of derivatives that can be taken for such functions. In multivariable calculus, you will deal with partial derivatives. A partial derivative measures how a function changes as only one of the variables changes, while others are held constant. This concept is crucial in fields that require analyzing systems with multiple varying factors.

Understanding this concept is key not only in academics but also in practical applications like physics, economics, and engineering. You often deal with situations where multiple inputs or factors lead to an output result. Being able to analyze how one of those inputs affects the outcome while keeping others constant offers deep insights.

When studying multivariable functions, always remember:

• Identify the variables and the dependent relationship between them.
• Understand how changing one variable while holding others constant influences the function.
• Interpretation of results depends on the context of the problem.

Differentiation

Differentiation is a fundamental concept in calculus, and when it comes to functions of multiple variables, the process requires focusing on one variable at a time. This is where partial derivatives come in.

To compute the partial derivative of a function like \( z = f(x, y) \), you treat one of the variables, say \( y\), as a constant and differentiate with respect to \(x\). This process is reversed to find the derivative with regard to \( y \), treating \( x \) as constant.

Let's go through an example using the function \( z = 5x + 11y \):

• For the partial derivative with respect to \( x \), denoted as \( \frac{\partial z}{\partial x} \), you treat \( y \) as a constant. Applying basic differentiation rules, \( \frac{\partial}{\partial x}(5x + 11y) = 5 \).
• For the partial derivative with respect to \( y \), denoted as \( \frac{\partial z}{\partial y} \), you treat \( x \) as a constant. Applying differentiation rules, \( \frac{\partial}{\partial y}(5x + 11y) = 11 \).

Differentiation helps to establish how sensitive a function is to changes in each of its inputs, providing detailed insight into the function's structure. It becomes incredibly useful when optimizing functions and solving complex real-world problems.

Engineering Mathematics

Engineering mathematics heavily relies on the principles of multivariable calculus and differentiation. Engineers use these concepts to model and analyze physical systems in which several factors influence the output.

For example, in thermodynamics, engineers might model a system where the pressure, volume, and temperature simultaneously impact the state of a gas. Understanding the partial derivatives helps engineers predict how a change in one parameter affects the others.

By applying math from calculus:

• Engineers can calculate stress and strain in structural elements by considering how forces distributed in different directions affect the element.
• Electrical engineers analyze circuits, understanding how changes in voltage or resistance impact current flow with respect to one part of the circuit.
• In mechanics, predicting how a change in one dimension affects the overall mechanics of an object is often required.

These mathematical principles give engineers the tools to create, simulate, and improve designs, ensuring safety and efficiency in technology and structures. It’s the backbone of tangible, practical applications derived from theoretical mathematics.