Partial Derivatives and Implicit Differentiation

The video tutorial explains how to find the partial derivatives of a function involving variables x, y, and z using the implicit function theorem. It demonstrates the process step-by-step, first by moving terms to one side of the equation and then calculating the derivatives with respect to x and y. An alternative method using implicit differentiation is also presented. The tutorial concludes with a practice example to reinforce the concepts learned.

What is the first step in finding the partial derivative of z with respect to x using the implicit function theorem?

• Solve for z

• Set all terms to zero

• Differentiate with respect to y

• Move the constant to the other side of the equation

When finding the partial derivative of z with respect to x, what is the derivative of x squared?

• 2x

• x^2

• 0

• x

In the implicit function theorem, what is the expression for f sub z?

• 3z^2

• -3z^2

• z^3

• -z^3

What is the alternative method introduced for finding partial derivatives?

• Explicit differentiation

• Implicit differentiation

• Graphical differentiation

• Numerical differentiation

In implicit differentiation, what do you do when differentiating a z variable?

• Subtract the partial derivative of z with respect to y

• Ignore the z variable

• Multiply by the derivative of x

• Add the partial derivative of z with respect to x

What is the result of differentiating y to the fourth with respect to y?

• 3y^2

• 0

• y^4

• 4y^3

In the practice example, what is the partial derivative of z with respect to x?

• 2x^2z

• 3xy^2

• y^3 + 2z^2

• x^2 + y^3

What is the partial derivative of z with respect to y in the practice example?

• 2x^2z

• x^2 + y^3

• y^3 + 2z^2

• 3xy^2

How do you simplify the expression for the partial derivative of z with respect to y?

• Add a y variable

• Cancel an x variable

• Subtract a constant

• Multiply by z

What is the final expression for the partial derivative of z with respect to y in the practice example?

• -3y^2 / 2xz

• 3xy^2 / 2xz

• y^3 / xz

• 2x^2z / y^2