Partial Derivatives and Mixed Derivatives

This video tutorial covers the concept of partial derivatives, including how to find them with respect to different variables. It explains the power rule, derivatives of exponential functions, natural logarithms, square roots, and trigonometric functions. The tutorial also demonstrates evaluating derivatives at specific points and introduces higher order partial derivatives.

What is the partial derivative of the function f(x, y) = 7x^2 - x^3y^4 + 5x^4y^3 with respect to x?

• 7x^2 - 3x^3y^4 + 5x^4y^3

• 7x - 3x^2y^3 + 5x^4y^2

• 14x^2 - 3xy^4 + 20x^3y^2

• 14x - 3x^2y^4 + 20x^3y^3

How do you find the partial derivative of an exponential function e^(x^2y) with respect to x?

• e^(x^2y) * 2xy

• e^(x^2y) * 2x

• e^(x^2y) * y

• e^(x^2y) * x

What is the partial derivative of ln(x^2 + y^2) with respect to x?

• y / (x^2 + y^2)

• 2y / (x^2 + y^2)

• x / (x^2 + y^2)

• 2x / (x^2 + y^2)

When evaluating the function f(x, y) = 2x^3y^2 + 5y + 4x^2 at the point (1, 2), what is the value of the partial derivative with respect to x?

• 16

• 24

• 8

• 32

In the function f(x, y) = x^3e^(4xy^2), when should you use the product rule to find the partial derivative with respect to x?

• When only one part of the function contains x

• When the function is a constant

• When both parts of the function contain x

• When neither part of the function contains x

What is the second partial derivative of f(x, y) = x^3 + 4x^5y^3 + 5y^4 with respect to x?

• 6x + 80x^3y^3

• 3x + 80x^4y^2

• 6x^2 + 20x^3y^2

• 3x^2 + 20x^4y^3

How do you find the mixed partial derivative f(x, y, z) = x^3y^4 + z^2x^3y with respect to x, y, and then z?

• 6x^2

• 6x^2y

• 6x^2z

• 6xyz

What does the equality of mixed partial derivatives theorem state?

• Mixed partial derivatives are always zero

• The order of differentiation always matters

• Mixed partial derivatives are always equal to the original function

• The order of differentiation does not matter if the function is continuous

If f(x, y) = x^2y^3 + 4xy^2, what is the mixed partial derivative fxy?

• 6xy^2 + 8y

• 3x^2y^2 + 8xy

• 6x^2y + 8

• 3xy^2 + 4

For the function f(x, y, z) = 4x^2y^3z + 6xy^2z^2, what is f(x, y, y)?

• 24xy^2z + 12

• 48xyz + 12z^2

• 24x^2yz + 12z

• 48x^2y + 12z^2