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Chapter 13: Problem 13

Find both first partial derivatives. $$ z=\ln \left(x^{2}+y^{2}\right) $$

Short Answer

Expert verified
The first order partial derivatives are \(\frac{\partial z}{\partial x}=\frac{2x}{x^2+y^2}\) and \(\frac{\partial z}{\partial y}=\frac{2y}{x^2+y^2}\).

Step by step solution

01

Partial Derivative with Respect to x

First, consider only \(x\) as the variable and \(y\) as constant. The partial derivative of \(z\) with respect to \(x\) then is: \[ \frac{\partial z}{\partial x}=\frac{1}{x^{2}+y^{2}} \cdot 2x \] Simplifying, we get: \[ \frac{\partial z}{\partial x}=\frac{2x}{x^2+y^2} \]
02

Partial Derivative with Respect to y

Next, consider only \(y\) as the variable and \(x\) as constant. The partial derivative of \(z\) with respect to \(y\) is then: \[ \frac{\partial z}{\partial y}=\frac{1}{x^{2}+y^{2}} \cdot 2y \] Simplifying, we get: \[ \frac{\partial z}{\partial y}=\frac{2y}{x^2+y^2} \]

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Key Concepts

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

Calculus
If you're new to calculus, think of it as a mathematical tool that enables us to analyze changes. At its heart lies the concept of a derivative, which measures how a function changes as its input changes. In the given exercise, we are dealing with partial derivatives, which are simply the derivatives of functions with more than one variable. In practical terms, it's like we're examining how a surface (represented by a function of two variables, in this case) slopes if we move along one axis at a time, holding the other constant.
Multivariable Calculus
We're stepping into the realm of multivariable calculus when we deal with functions like the one in our exercise, which depends on more than one input. Here, the function's output, denoted as \(z\), depends on both \(x\) and \(y\). Multivariable calculus allows us to extend the principles of single-variable calculus, like differentiation and integration, into higher dimensions. Practical applications of multivariable calculus range from physics and engineering to economics and biology, where phenomena are often dependent on several factors.
Chain Rule
The chain rule is a formula that helps us tackle the derivatives of composite functions. When functions are tangled together, the chain rule tells us how to find the derivative of this mix step by step. We used the chain rule implicitly when finding the partial derivatives in our exercise. We treated \(y\) as a constant when differentiating with respect to \(x\) and vice versa. In more complicated situations, the chain rule would guide us through the differentiation by systematically taking apart the functions layer by layer, like peeling an onion.
Natural Logarithm
Natural logarithm (ln) is a special function that's intimately connected to the notion of e, the base of the natural logarithm, which is approximately equal to 2.71828. In calculus, the natural logarithm is notable for being the inverse of the exponential function with base e. This means that it 'undoes' the exponential function. In our specific problem, \(z = \ln(x^2 + y^2)\), the natural logarithm helps us model scenarios with growth patterns or tackle equations involving exponentials. Its properties, such as \(\ln(ab) = \ln(a) + \ln(b)\), are frequently utilized to simplify calculus expressions.

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Most popular questions from this chapter

Use Lagrange multipliers to find the minimum distance from the curve or surface to the indicated point. [Hint: In Exercise 23, minimize \(f(x, y)=x^{2}+y^{2}\) subject to the constraint \(2 x+3 y=-1 .]\) $$ \begin{array}{ll} \underline{\text { Curve }} & \underline{\text {Point}} \\ \text { Line: } 2 x+3 y=-1 \quad (0,0) \end{array} $$

For the function \(f(x, y)=x y\left(\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\right)\) define \(f(0,0)\) such that \(f\) is continuous at the origin.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a cylindrical surface \(z=f(x, y)\) has rulings parallel to the \(y\) -axis, then \(\partial z / \partial y=0\)

Use Lagrange multipliers to find the indicated extrema of \(f\) subject to two constraints. In each case, assume that \(x, y\), and \(z\) are nonnegative. Maximize \(f(x, y, z)=x y z\) Constraints: \(x^{2}+z^{2}=5, \quad x-2 y=0\)

Explain the Method of Lagrange Multipliers for solving constrained optimization problems.
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