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Chapter 11: Problem 6

Find both first partial derivatives. \(f(x, y)=x^{2}-3 y^{2}+7\)

Short Answer

Expert verified
The first partial derivatives of the function are \( \partial f/ \partial x = 2x \) and \( \partial f/ \partial y = -6y \).

Step by step solution

01

Find the partial derivative with respect to x

To find the partial derivative of \(f\) with respect to \(x\), treat all other variables as constants. The derivative of the constant \(7\) drops out, and the derivative of the term \(-3y^2\) is 0 because \(y\) is treated as a constant. This yields: \( \partial f/ \partial x = 2x \).
02

Find the partial derivative with respect to y

Similarly, to find the partial derivative of \(f\) with respect to \(y\), treat all other variables as constants. The \(x^2\) term yields 0 because \(x\) is treated as a constant, and the \(7\) term drops out. This yields: \( \partial f/ \partial y = -6y \).

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Key Concepts

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

Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Its development in the 17th century was primarily driven by the needs to understand motion and change in the physical world. It's split into two main areas: differential calculus and integral calculus. Differential calculus concerns the determination of the instant rate of change of quantities, which is essentially what we know as the derivative of a function.

The concept of the derivative is fundamental in calculus and has vast applications, including solving problems in physics, engineering, economics, and beyond. It allows for the calculation of how a function changes at any given point and is crucial for tasks like finding maximum and minimum values of functions.
Multivariable Calculus
Multivariable calculus is the extension of calculus in one variable to calculus with functions of several variables: the differentiation and integration of functions involving multiple variables, rather than just one.

Consider a function such as the elevation of a mountain, which depends on the two geographical coordinates: latitude and longitude. In multivariable calculus, we consider such functions and analyze their behavior. This discipline also delves into more advanced topics such as vector calculus, partial derivatives, multiple integrals, and vector fields. Multivariable calculus is essential in fields like physics, engineering, and economics, where systems depend on more than one variable.
First Partial Derivative
The first partial derivative of a multivariable function is the derivative with respect to one variable while holding the other variables constant. To illustrate, let us consider a function like the one from our exercise, which is a function of two variables, x and y.

When we calculate the partial derivative of this function with respect to x, denoted as \( \partial f/\partial x \), we treat y as if it were a constant, which means any terms not involving x are treated as constant and their derivatives equate to zero. This process is mirrored when finding \( \partial f/ \partial y \) by treating x as a constant.

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Most popular questions from this chapter

Use the function to prove that (a) \(f_{x}(0,0)\) and \(f_{y}(\mathbf{0}, \mathbf{0})\) exist, and (b) \(f\) is not differentiable at \((\mathbf{0}, \mathbf{0})\). \(f(x, y)=\left\\{\begin{array}{ll}\frac{5 x^{2} y}{x^{3}+y^{3}}, & (x, y) \neq(0,0) \\ 0, & (x, y)=(0,0)\end{array}\right.\)

The temperature at the point \((x, y)\) on a metal plate is \(T=\frac{x}{x^{2}+y^{2}}\). Find the direction of greatest increase in heat from the point (3,4) .

Find the gradient of the function and the maximum value of the directional derivative at the given point. $$ \frac{\text { Function }}{f(x, y, z)=\sqrt{x^{2}+y^{2}+z^{2}}} \frac{\text { Point }}{(1,4,2)} $$

Show that the function is differentiable by finding values for \(\varepsilon_{1}\) and \(\varepsilon_{2}\) as designated in the definition of differentiability, and verify that both \(\varepsilon_{1}\) and \(\varepsilon_{2} \rightarrow 0\) as \((\boldsymbol{\Delta x}, \boldsymbol{\Delta} \boldsymbol{y}) \rightarrow(\mathbf{0}, \mathbf{0})\) \(f(x, y)=x^{2}+y^{2}\)

In Exercises \(63-66,\) the function \(f\) is homogeneous of degree \(n\) if \(f(t x, t y)=t^{n} f(x, y) .\) Determine the degree of the homogeneous function, and show that \(x f_{x}(x, y)+y f_{y}(x, y)=n f(x, y)\) \(f(x, y)=\frac{x y}{\sqrt{x^{2}+y^{2}}}\)
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