Question

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

Short answer

The first partial derivatives of the function \(w = xyz\) with respect to \(x\), \(y\), and \(z\) are \(\frac{\partial w}{\partial x} = yz\), \(\frac{\partial w}{\partial y} = xz\), and \(\frac{\partial w}{\partial z} = xy\), respectively.

Step-by-step solution

Finding the Partial Derivative with respect to x

Differentiate the function \(w = xyz\), with respect to \(x\), while keeping \(y\) and \(z\) as constants. This gives: \(\frac{\partial w}{\partial x} = yz\)

Finding the Partial Derivative with respect to y

Next, differentiate \(w = xyz\) with respect to \(y\), keeping \(x\) and \(z\) as constants. This gives: \(\frac{\partial w}{\partial y} = xz\)

Finding the Partial Derivative with respect to z

Lastly, differentiate \(w = xyz\) with respect to \(z\), keeping \(x\) and \(y\) as constants. This leads to: \(\frac{\partial w}{\partial z} = xy\)

Key concepts

Multivariable Calculus

Multivariable calculus is an extension of single-variable calculus that includes functions of two or more variables. It involves concepts such as partial differentiation, multiple integrals, and vector calculus, which are essential in understanding the behavior of functions in higher-dimensional spaces.

In multivariable calculus, we study phenomena where several quantities change simultaneously and explore the relationships between these changing quantities. Applications of multivariable calculus can be found in various fields such as physics, engineering, and economics, where it is used to model and analyze complex systems with multiple interacting variables.

When we deal with functions of multiple variables, we can no longer talk about 'the' derivative, since there are many directions in which a function can change. Instead, we focus on partial derivatives, which measure how a function changes as one specific variable changes while holding the others constant.

A primary goal in multivariable calculus is understanding how these partial derivatives interact and combine to describe the overall behavior of a function, particularly through the lens of the gradient, which is a vector composed of all the first partial derivatives of a function.

First Partial Derivative

The first partial derivative of a function with respect to a given variable is the derivative taken with only that variable changing, while all other variables are held constant. Imagine you are hiking on a mountain, and you're interested in only how steep the incline is going directly north; that’s an analogy for a partial derivative — it's the rate of change in one specified direction.

In the exercise provided, finding the first partial derivatives of the multivariable function w = xyz with respect to each variable separately, we consider how w changes as only one of x, y, or z varies at a time. The resulting expressions, yz, xz, and xy represent how sensitive the function is to small changes in each of those variables individually.

These calculations are foundational in multivariable calculus as they provide critical information used to optimize processes, understand geometric properties of surfaces, and even in machine learning algorithms to find the steepest ascent or descent in multi-dimensional space.

Differentiation

Differentiation is a fundamental concept in calculus, used to determine the rate at which a quantity changes. In the context of single-variable calculus, differentiation focuses on how a function f(x) changes as the variable x changes. The derivative of f with respect to x, denoted as f'(x) or df/dx, is a way of encapsulating this rate of change at any point along the function’s domain.

In multivariable calculus, differentiation includes finding partial derivatives for each variable in a multi-dimensional function. This process involves treating all other variables as constants and applying the rules of differentiation to the variable of interest. Techniques from single-variable differentiation, such as the product rule and the chain rule, can also be extended to partial differentiation with slight modifications.

Understanding differentiation and its principles is vital because it allows us to solve a host of problems, from finding tangents to curves and determining instantaneous velocities, to predicting the future course of dynamic systems. The partial derivatives in the exercise are examples of how differentiation is applied in multivariable functions to find the tangent planes to surfaces and predict how a change in one variable affects the outcome when multiple factors are at play.