Problem 17 - APEX Calculus

Question

In Exercises \(17-18\), find the total differential \(d w\). $$ w=x^{2} y z^{3} $$

Step-by-Step Solution

Answer

The total differential is \(dw = 2x y z^3 \, dx + x^2 z^3 \, dy + 3x^2 y z^2 \, dz\).

Step 1: Identify Variables and Partial Derivatives

To find the total differential, identify the variables involved in the function. Here, the function is given as \(w = x^2 y z^3\). The variables are \(x\), \(y\), and \(z\). We need to calculate the partial derivatives: \(\frac{\partial w}{\partial x}, \frac{\partial w}{\partial y},\) and \(\frac{\partial w}{\partial z}\).

Step 2: Calculate Partial Derivative with Respect to x

Differentiate the function \(w = x^2 y z^3\) with respect to \(x\), treating \(y\) and \(z\) as constants. The partial derivative is \(\frac{\partial w}{\partial x} = 2x \, yz^3\).

Step 3: Calculate Partial Derivative with Respect to y

Differentiate the function \(w = x^2 y z^3\) with respect to \(y\), treating \(x\) and \(z\) as constants. The partial derivative is \(\frac{\partial w}{\partial y} = x^2 z^3\).

Step 4: Calculate Partial Derivative with Respect to z

Differentiate the function \(w = x^2 y z^3\) with respect to \(z\), treating \(x\) and \(y\) as constants. The partial derivative is \(\frac{\partial w}{\partial z} = 3x^2 y z^2\).

Step 5: Construct the Total Differential

Using the formula for the total differential, \(dw = \frac{\partial w}{\partial x} dx + \frac{\partial w}{\partial y} dy + \frac{\partial w}{\partial z} dz\), substitute the partial derivatives calculated:\[dw = (2x \, yz^3) dx + (x^2 z^3) dy + (3x^2 y z^2) dz.\]

Key Concepts

Partial Derivatives

Partial derivatives are essential in understanding how a function changes as one of its variables changes, while the others remain constant. In the context of multivariable functions like \( w = x^2 y z^3 \), partial derivatives are calculated with respect to each variable. For example, when finding \( \frac{\partial w}{\partial x} \), we treat \( y \) and \( z \) as constants and differentiate only with respect to \( x \). This shows how the function \( w \) changes as \( x \) varies, providing insight into the sensitivity of \( w \) to changes in \( x \). Similarly:

• \( \frac{\partial w}{\partial y} \) considers \( x \) and \( z \) as constants.
• \( \frac{\partial w}{\partial z} \) sees \( x \) and \( y \) as constants.

Partial derivatives are building blocks for understanding more complex changes in multivariable calculus, such as computing total differentials. They help us grasp the varying impacts of different independent variables on the function's outcome.

Multivariable Calculus

Multivariable calculus expands upon single-variable calculus by handling functions with more than one variable. It allows us to understand the behavior of systems influenced by multiple factors. In the given exercise, we deal with the function \( w = x^2 y z^3 \), involving three variables \( x \), \( y \), and \( z \). This opens many doors:

• Describing complex systems such as the motion of objects in three-dimensional space.
• Analyzing interactions across various fields, from physics to economics.
• Evaluating gradients, which indicate the direction of maximum increase of functions.

By using multivariable calculus, we can find total differentials like \( dw \). This is crucial for predicting how small changes in \( x \), \( y \), and \( z \) will affect \( w \). This technique is essential for applications in optimization and modeling real-world situations.

Differentiation

Differentiation is the process of finding a derivative, describing how a function changes as its inputs change. In multivariable calculus, this involves taking partial derivatives to evaluate how a function responds to changes in one of its variables. For a function \( w = f(x, y, z) \), we derive each partial derivative to understand the individual effect of each variable. This allows us to express the total differential:\[ dw = \frac{\partial w}{\partial x} \, dx + \frac{\partial w}{\partial y} \, dy + \frac{\partial w}{\partial z} \, dz \]This expression provides a comprehensive view of differentiation in multivariable contexts. Each term shows how changes in one variable, compounded infinitesimally, affect the entire function. Differentiation in this manner is pivotal in fields that require sensitivity analysis, such as engineering or finance. Understanding and applying differentiation help us approximate real-world phenomena accurately, aiding precise calculations and predictions.