Question

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

Short answer

\(\frac{\partial w}{\partial x} = \frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\), \(\frac{\partial w}{\partial y} = \frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\), \(\frac{\partial w}{\partial z} = \frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\)

Step-by-step solution

Partial Derivative with respect to \(x\)

To find the first partial derivative with respect to \(x\), hold the other variables, \(y\) and \(z\), constant and differentiate with respect to \(x\). Applying the chain rule, we have: \[\frac{\partial w}{\partial x} = \frac{1}{2\sqrt{x^{2}+y^{2}+z^{2}}} \cdot 2x = \frac{x}{\sqrt{x^{2}+y^{2}+z^{2}}}\]

Partial Derivative with respect to \(y\)

In a similar fashion, find the first partial derivative with respect to \(y\) by holding the other variables, \(x\) and \(z\), constant and differentiate with respect to \(y\). Using the chain rule, we obtain: \[\frac{\partial w}{\partial y} = \frac{1}{2\sqrt{x^{2}+y^{2}+z^{2}}} \cdot 2y = \frac{y}{\sqrt{x^{2}+y^{2}+z^{2}}}\]

Partial Derivative with respect to \(z\)

Finally, to find the first partial derivative with respect to \(z\), hold \(x\) and \(y\) constant and differentiate with respect to \(z\). Applying the chain rule again, we find: \[\frac{\partial w}{\partial z} = \frac{1}{2\sqrt{x^{2}+y^{2}+z^{2}}} \cdot 2z = \frac{z}{\sqrt{x^{2}+y^{2}+z^{2}}}\]

Key concepts

Multivariable Calculus

When diving into the realm of multivariable calculus, you encounter functions with more than one variable — such as the function given in the exercise, where the output, labeled as w, depends on three different inputs: x, y, and z.

Understanding how to work with such functions is crucial to various fields like physics, engineering, and economics. One of the primary tools in multivariable calculus is the partial derivative, which measures how a function changes as one of the variables is varied while the others are held fixed.

Partial derivatives are represented as ∂w/∂x, ∂w/∂y, and ∂w/∂z in our example. They reflect the rate of change of the function with respect to one variable at a time, giving us insight into the function’s behavior along various axes in its domain.

Chain Rule

The chain rule is a cornerstone concept of calculus, both single-variable and multivariable. It allows us to tackle the differentiation of composite functions — functions that involve other functions. In the exercise, the function w = √(x² + y² + z²) is a composition of the square root function and the sum of squares.

In simpler terms, if you consider a situation where changes in one quantity cause changes in a second quantity, which in turn cause changes in a third quantity, the chain rule helps you understand the total effect of these changes. For instance, in the provided step-by-step solution, when calculating ∂w/∂x, the chain rule is employed to connect the derivative of the outer square root function to the derivative of the inner function x².

It's pivotal to correctly identify the 'inner' and 'outer' functions when applying the chain rule to ensure the differentiation is performed correctly. This concept is invaluable when dealing with more complex structures encountered in higher-level math and practical applications.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function's output changes with respect to changes in its input. It's a fundamental tool in calculus used to analyze the behavior of functions.

In the context of multivariable functions, differentiation becomes slightly more intricate. Each partial derivative focuses on the change along one axis, temporarily ignoring the others. This gives a directional sense of how the function behaves — similar to how you might study the features of a landscape by looking at how the terrain slopes in various directions.

In practicing differentiation, especially in the case of multivariable calculus, recognizing the type of function you're dealing with can significantly streamline the process. For example, functions involving roots often require the use of the chain rule, as seen in the solution for w. Differentiation is widely applied in optimizing systems, modeling natural phenomena, and in the theoretical foundations of the physical sciences and economics.