Partial Derivatives in Implicit Functions

The video tutorial explains how to calculate the partial derivatives of z with respect to x and y from an implicit equation set to zero. It introduces the concept of implicit equations and partial derivatives, then demonstrates the calculation of these derivatives at a specific point using the function F(x, y, z). The tutorial provides step-by-step differentiation of the function with respect to x and y, treating other variables as constants, and evaluates the derivatives at the given point.

What is the form of the implicit equation used in the problem?

• f(x, y, z) = 0

• f(y, z) = x

• f(x, y) = z

• f(x, z) = y

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

• By dividing the partial of f with respect to x by the partial of f with respect to y

• By dividing the partial of f with respect to z by the partial of f with respect to x

• By dividing the partial of f with respect to x by the partial of f with respect to z

• By dividing the partial of f with respect to y by the partial of f with respect to z

What is the derivative of 3x^3 with respect to x?

• 6x^2

• 12x^2

• 3x^2

• 9x^2

What is the value of the partial derivative of z with respect to x at the point (2, -3, 6)?

• 1/67

• 9/67

• -9/67

• -1/67

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

• 5xz

• 5xy

• 5yz^2

• 5yz

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

• By dividing the partial of f with respect to y by the partial of f with respect to x

• By dividing the partial of f with respect to z by the partial of f with respect to y

• By dividing the partial of f with respect to y by the partial of f with respect to z

• By dividing the partial of f with respect to x by the partial of f with respect to z

What is the derivative of 2y^3 with respect to y?

• 3y^2

• -3y^2

• 6y^2

• -6y^2

What is the value of the partial derivative of z with respect to y at the point (2, -3, 6)?

• 1/67

• -1/67

• 9/67

• -9/67

What is the derivative of 4z^3 with respect to z?

• 16z^2

• 12z^2

• 20z^2

• 8z^2

What is the derivative of 5xyz with respect to y?

• 5xz

• 5xz^2

• 5yz

• 5xy