Problem 120 - Calculus 3

Question

For the following exercises, calculate the partial derivatives. \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\) for \(z=x^{8} e^{3 y}\)

Step-by-Step Solution

Answer

\(\frac{\partial z}{\partial x} = 8x^7 e^{3y}\), \(\frac{\partial z}{\partial y} = 3x^8 e^{3y}\).

Step 1: Identify the Function and Variables

We are given the function \(z = x^8 e^{3y}\). We need to find the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). In this case, \(z\) is the dependent variable, and \(x\) and \(y\) are the independent variables.

Step 2: Calculate \(\frac{\partial z}{\partial x}\)

To find the partial derivative of \(z\) with respect to \(x\), we treat \(y\) as a constant. Therefore, we only differentiate the part of the function that involves \(x\), which is \(x^8\). Using the power rule, the derivative of \(x^8\) with respect to \(x\) is \(8x^7\). So, \(\frac{\partial z}{\partial x} = 8x^7 e^{3y}\).

Step 3: Calculate \(\frac{\partial z}{\partial y}\)

To find the partial derivative of \(z\) with respect to \(y\), we treat \(x\) as a constant. So, we differentiate the part of the function that involves \(y\), which is \(e^{3y}\). Using the chain rule, the derivative of \(e^{3y}\) is \(3e^{3y}\). So, \(\frac{\partial z}{\partial y} = x^8 \cdot 3e^{3y} = 3x^8 e^{3y}\).

Key Concepts

Calculating Derivatives

When learning to calculate derivatives, it's essential to understand that a derivative represents the rate of change of a function. In multivariable calculus, we deal with functions that have more than one variable. Instead of a single slope, we have partial derivatives, which measure the rate of change of a function concerning one variable while keeping other variables constant. The primary aim in calculating derivatives is to find these rates of change, often using different rules like the power rule and the chain rule, which simplify the differentiation process.

Power Rule

The Power Rule is a fundamental tool in calculus for finding derivatives. It states that if you have a function in the form of \( x^n \), where \( n \) is a constant, the derivative with respect to \( x \) is \( nx^{n-1} \).

• For example, if \( z = x^8 \), the derivative would be \( 8x^7 \).
• This rule makes finding derivatives of polynomial terms straightforward and is especially useful when handling functions in multivariable calculus where one or more variables might be involved.

Understanding and correctly applying the power rule lets you solve complex functions more efficiently.

Chain Rule

The Chain Rule is another essential calculus tool that helps find derivatives of composite functions. A composite function occurs when one function is nested inside another, like \( e^{3y} \) in our example problem.

• To use the chain rule, differentiate the outer function while keeping the inner function unchanged and then multiply by the derivative of the inner function.
• In the case of \( e^{3y} \), differentiate the outer function \( e^u \) to get \( e^u \), and multiply it by the derivative of \( 3y \), which is 3. This yields \( 3e^{3y} \).

Applying the chain rule correctly allows you to tackle more complex functions without breaking them down into less workable parts.

Multivariable Calculus

Multivariable calculus extends single-variable calculus concepts to functions with several variables. Here, it's crucial to understand partial derivatives, which tell us how a function changes as one variable changes while others are held constant.

• In the given exercise, \( z = x^8 e^{3y} \), calculate \( \frac{\partial z}{\partial x} \) by considering \( y \) as a constant. This results in applying the power rule solely to \( x^8 \).
• Similarly, calculate \( \frac{\partial z}{\partial y} \) by keeping \( x \) constant and using the chain rule for the exponential term \( e^{3y} \).

Multivariable calculus is invaluable for understanding and modeling realistically complex phenomena involving more than one changing variable.