Partial Derivatives and Their Rules

The video tutorial explains how to find the first-order partial derivatives of a function involving variables X, Y, and Z. It begins by rewriting the function using a rational exponent. The tutorial then demonstrates how to calculate the partial derivatives with respect to each variable, treating the other variables as constants. The process involves applying the power rule and chain rule, and simplifying the results to express them in terms of square roots. Each step is explained in detail, ensuring a clear understanding of the derivation process.

What is the first step in finding the partial derivatives of the given function?

• Differentiate with respect to all variables simultaneously.

• Apply the chain rule directly.

• Use the product rule.

• Rewrite the function using a rational exponent.

When finding the partial derivative with respect to X, which terms are treated as constants?

• X and Z

• Y and Z

• None of the terms

• X and Y

What rule is applied along with the power rule to find the partial derivative with respect to X?

• Chain rule

• Quotient rule

• Product rule

• Sum rule

What is the derivative of 5x with respect to X?

• 6

• 4

• 5

• 0

In finding the partial derivative with respect to Y, what is the derivative of 6y with respect to Y?

• 0

• 5

• 4

• 6

What happens to the exponent when moving the term to the denominator in the partial derivative with respect to Y?

• It doubles.

• It becomes positive.

• It halves.

• It becomes negative.

When finding the partial derivative with respect to Z, which terms are treated as constants?

• X and Z

• Y and Z

• X and Y

• None of the terms

What is the derivative of -4Z with respect to Z?

• 0

• 6

• 4

• -4

What is the result of multiplying 12 by -4 in the context of the partial derivative with respect to Z?

• 2

• -2

• 48

• -48

What is the final form of the denominator in the partial derivative with respect to Z?

• A quadratic term

• A linear term

• A square root

• A cube root