Problem 123 - Calculus 3

Question

For the following exercises, calculate the partial derivatives. Let \(z=e^{x y} .\) Find \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\)

Step-by-Step Solution

Answer

\(\frac{\partial z}{\partial x} = ye^{xy}, \; \frac{\partial z}{\partial y} = xe^{xy}\).

Step 1: Understanding the Partial Derivative with Respect to x

We start by understanding the function given: \( z = e^{xy} \). To find the partial derivative of \( z \) with respect to \( x \), we treat \( y \) as a constant. This means we differentiate \( e^{xy} \) as a function of \( x \) only.

Step 2: Applying the Chain Rule for \(\frac{\partial z}{\partial x}\)

Using the chain rule, we find the derivative of the exponent \(xy\) with respect to \(x\), which is \(y\) (since \(y\) is constant). Then, the derivative of \( e^{xy} \) with respect to \(x\) is \( y \cdot e^{xy} \). Thus, \(\frac{\partial z}{\partial x} = ye^{xy}\).

Step 3: Understanding the Partial Derivative with Respect to y

Next, we find the partial derivative of \( z \) with respect to \( y \). We treat \( x \) as a constant in this case, meaning we view \( e^{xy} \) in terms solely of \( y \).

Step 4: Applying the Chain Rule for \(\frac{\partial z}{\partial y}\)

Compute the derivative of the exponent \(xy\) with respect to \(y\), which is \(x\) (since \(x\) is constant now). Therefore, the derivative of \( e^{xy} \) with respect to \(y\) is \( x \cdot e^{xy} \). Thus, \(\frac{\partial z}{\partial y} = xe^{xy}\).

Key Concepts

Chain Rule

When dealing with complex functions, such as exponentials, we frequently need to use the chain rule. The chain rule is a fundamental technique in calculus for differentiating composite functions. This rule helps us find derivatives for functions that are built from other simpler functions.

For example, if we have a function like \( z = e^{xy} \), the inside piece is \( xy \), and the outside is the exponential function \( e^u \), where \( u = xy \). To differentiate \( e^{xy} \), we need to:

• First, differentiate the outside function (exponential) with respect to its argument. The derivative of \( e^u \) is simply \( e^u \).
• Then, multiply this by the derivative of the inside function \( xy \) with respect to the variable of interest (\( x \) or \( y \) here).

This results in derivatives like \( y \cdot e^{xy} \) or \( x \cdot e^{xy} \) depending on whether we differentiate with respect to \( x \) or \( y \), respectively. The chain rule efficiently handles these layers of complexity.

Multivariable Calculus

In multivariable calculus, we work with functions that depend on more than one variable. This expands our capabilities from single-variable calculus, allowing us to explore space in new ways.

For the function \( z = e^{xy} \), understanding the behavior requires knowing how \( z \) changes with respect to both \( x \) and \( y \). This is where the concept of partial derivatives comes in.

When computing the partial derivative with respect to \( x \), \( y \) is treated as a constant. And when computing with respect to \( y \), \( x \) is treated as constant. Thus:

• Partial derivatives give us the rate of change of \( z \) in each direction, individually.
• This gives a more comprehensive view of how \( z \) behaves as the input variables change.

Multivariable calculus allows us to see and predict how small changes in \( x \) or \( y \) separately lead to changes in the function \( z \). It is like looking at how the surface of \( z \) slopes in different directions.

Exponential Function Differentiation

Exponential functions are prevalent in calculus due to their unique properties. Differentiating these functions can sometimes seem tricky, but they follow some straightforward rules.

Consider \( z = e^{xy} \). We apply exponential differentiation as follows:

• The derivative of an exponential function \( e^u \) (where \( u \) is a function of one or more variables) is \( e^u \) itself.
• When the exponent is a product of variables, like \( xy \), we use the chain rule to find derivatives with respect to each variable.

For the partial derivative with respect to \( x \), the step involves differentiating the inside function \( xy \) to obtain \( y \). Multiply by the whole exponent \( e^{xy} \) to get \( ye^{xy} \).

With exponential functions, these steps provide a methodical path to finding derivatives that yield powerful and concise results. These functions' properties help maintain the same form, making them easier to work with in calculus operations.