Partial Derivatives and Their Rules

This video tutorial covers the basics of partial derivatives, including notation and solving problems involving functions, exponential and logarithmic functions. It also tackles an advanced problem involving third partial derivatives, providing step-by-step solutions and explanations.

What is the notation for the partial derivative of a function f with respect to x?

• dx/df

• df/dx

• ∂f/∂x

• ∂x/∂f

In the context of partial derivatives, what does it mean to keep a variable constant?

• Differentiate the variable normally

• Set the variable to zero

• Treat the variable as a constant value

• Ignore the variable completely

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

• y

• x^2

• 2xy

• 2x

When taking the partial derivative of a function with respect to y, what should be done with x?

• Treat x as a constant

• Differentiate x normally

• Ignore x

• Set x to zero

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

• e^(x+y)

• e^(x+y) * (x+y)

• e^(x+y) * y

• e^(x+y) * 1

Which rule is used when differentiating e^(x+y) with respect to x?

• Power rule

• Chain rule

• Quotient rule

• Product rule

What is the derivative of log base 3 of y^2 - x^2 with respect to x?

• -2x / ((y^2 - x^2) * ln(3))

• 2x / ((y^2 - x^2) * ln(3))

• -2x / ln(3)

• 2x / ln(3)

What is the partial derivative of log base 3 of y^2 - x^2 with respect to y?

• 2y / ((y^2 - x^2) * ln(3))

• -2y / ((y^2 - x^2) * ln(3))

• 2y / ln(3)

• -2y / ln(3)

What is the third partial derivative of a function with respect to x and y?

• The derivative of the first partial derivative

• The derivative of the second partial derivative

• The derivative of the derivative of the derivative

• The derivative of the original function

In the context of third partial derivatives, what does it mean to take the derivative with respect to y?

• Ignore y

• Set y to zero

• Differentiate y normally

• Treat y as a constant