Question

Find the first partial derivatives with respect to \(x, y\), and \(z\). $$ w=\frac{2 z}{x+y} $$

Short answer

The first order partial derivatives of \( w \) with respect to \(x, y, z\) are \(-\frac{2z}{(x + y)^2}\), \(-\frac{2z}{(x + y)^2}\), and \(\frac{2}{x+y}\) respectively.

Step-by-step solution

Compute Derivative with respect to \(x\)

Apply the quotient rule for derivatives \( \frac{d}{dx}[u/v] = \frac{vu' - uv'}{v^2} \) with \(u = 2z\) and \(v = x + y\). The derivative of \(u\) with respect to \(x\) is 0 and the derivative of \(v\) with respect to \(x\) is 1. Then, compute \( \frac{\partial w}{\partial x} = \frac{(x+y)*0 - 2z*1}{(x + y)^2} = -\frac{2z}{(x + y)^2} \)

Compute Derivative with respect to \(y\)

Similarly, for \( \frac{\partial w}{\partial y} \), apply the quotient rule with \(u = 2z\) and \(v = x + y\). The derivative of \(u\) with respect to \(y\) is 0 and the derivative of \(v\) with respect to \(y\) is 1. Then, compute \( \frac{\partial w}{\partial y} = \frac{(x+y)*0 - 2z*1}{(x + y)^2} = -\frac{2z}{(x + y)^2}\)

Compute Derivative with respect to \(z\)

For \( \frac{\partial w}{\partial z} \), apply the quotient rule with \(u = 2z\) and \(v = x + y\). The derivative of \(u\) with respect to \(z\) is 2 and the derivative of \(v\) with respect to \(z\) is 0. Then, compute \( \frac{\partial w}{\partial z} = \frac{(x+y)*2 - 2z*0}{(x + y)^2} = \frac{2(x + y)}{(x + y)^2} = \frac{2}{x + y} \)

Key concepts

Quotient Rule for Derivatives

Understanding the quotient rule for derivatives is crucial when dealing with functions that involve division. It is a method used in calculus to find the derivative of a ratio of two differentiable functions. Suppose we have a function written as a fraction where the numerator is u(x) and the denominator is v(x), the quotient rule states that the derivative of this function is given by:
\[ \frac{d}{dx}\left(\frac{u(x)}{v(x)}\right) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \]
In simpler terms, you multiply the denominator by the derivative of the numerator, and then subtract the product of the numerator and the derivative of the denominator, all over the square of the denominator. The rule is often remembered by the mnemonic 'low-dee-high minus high-dee-low over the square of what's below'.

First Partial Derivative

In multivariable calculus, the first partial derivative represents the rate at which a multivariable function changes as one of its variables is altered, while the others are held constant. More formally, given a function \( f(x, y, z, \dots) \), the first partial derivative with respect to \( x \) is denoted as \( \frac{\partial f}{\partial x} \) and is found by differentiating \( f \) with respect to \( x \) while treating \( y, z, \dots \) as constants. Partial derivatives are foundational in the study of multivariable functions, as they provide insight into a function's behavior with respect to each individual variable.

Calculus

Calculus, at its most basic level, is the mathematical study of change. It provides a framework for modeling systems in which there is change, and a way to deduce the predictions of such models. There are two major branches: differential calculus surrounds the concept of the derivative, which quantifies the rate of change of a function, and integral calculus, which measures the size of the area under or between curves. Both are interconnected and serve as the basis for a broad range of scientific disciplines, including physics, engineering, and economics.

Multivariable Calculus

While basic calculus deals with functions of a single variable, multivariable calculus extends these concepts to functions of multiple variables. This area of mathematics is not just about finding scalar derivatives of multivariable functions but also involves vectors and the study of vector fields. Here, the derivative takes the form of partial derivatives, and the integral extends into multiple integrals. Aspects such as gradient, divergence, and curl also come into play, providing detailed information about the behavior of three-dimensional scalar and vector fields. Understanding multivariable calculus is key in many areas, including physics, engineering, and computer graphics.