Understanding Partial Derivatives and the Chain Rule

The video tutorial explains how to use the chain rule to find the derivative dz/dt by employing partial derivatives. It covers the process of finding partial derivatives of z with respect to x and y, and then extends the concept to functions of multiple variables, including s and t. The tutorial also provides an example involving three variables, demonstrating the application of the chain rule in more complex scenarios.

What is the first step in using the chain rule to find dz/dt?

• Differentiate z with respect to x and y simultaneously.

• Find the derivative of z with respect to t directly.

• Identify the dependencies of z on x and y, and x and y on t.

• Use the product rule on z.

When finding the partial derivative of z with respect to x, what should y be treated as?

• An independent variable

• A function of t

• A constant

• A variable

What is the derivative of x with respect to t if x = 4 + t^2?

• t^2

• 4t

• 2t

• 4 + 2t

In the new problem, what are x and y functions of?

• Only t

• Neither s nor t

• Only s

• Both s and t

How do you find the partial derivative of z with respect to s?

• By differentiating z directly with respect to s.

• By treating s as a constant.

• By differentiating z with respect to x and y, then x and y with respect to s.

• By using the product rule on z.

What is the partial derivative of x with respect to t if x = s^2t?

• 3s^2

• 2st

• t^2

• s^2

In the complex example, what variables does u depend on?

• x, y, and z

• r, s, and t

• x, y, and t

• s, t, and z

What is the partial derivative of y with respect to t if y = r^2s*t^3?

• 3r^2s*t^2

• r^2s

• 2r^2s*t

• 3t^2

What is the final step in finding the partial derivative of u with respect to t?

• Multiply all partial derivatives together.

• Add the products of partial derivatives of u with respect to x, y, and z, and their respective derivatives with respect to t.

• Use the quotient rule on u.

• Differentiate u directly with respect to t.

What is the derivative of z with respect to t if z = rs*t^2?

• 2rs*t

• rs

• t^2

• 2t