Partial Derivatives and Chain Rule

The video tutorial explains how to find the partial derivative of the function 3xE^(3xy) with respect to x. It begins by treating y as a constant and differentiating with respect to x. The function is rewritten as F(x, y) = 3xE^(3xy), and the chain rule is applied, identifying the inner function U as 3xy. The derivative of U with respect to x is calculated as 3y. The partial derivative is then expressed as 9yE^(3xy), representing the slope of the tangent line in the x direction. The tutorial concludes with a brief application of the derivative.

What is the initial function given for finding the partial derivative with respect to x?

• 3y * e^(3xy)

• 3x * e^(xy)

• 3x * e^(3xy)

• 3y * e^(xy)

When finding the partial derivative with respect to x, how is y treated?

• As a constant

• As a variable

• As a function

• As a derivative

What is identified as the inner function u when applying the chain rule?

• 3y

• e^(3xy)

• 3xy

• 3x

What is the derivative of the inner function u = 3xy with respect to x?

• 3x

• 3

• 3y

• xy

What is the simplified expression for the partial derivative with respect to x?

• 3y * e^(3xy)

• 9x * e^(3xy)

• 9y * e^(3xy)

• 3x * e^(3xy)

What does the partial derivative with respect to x represent in terms of the function's graph?

• The maximum point of the graph

• The curvature of the graph

• The slope of the tangent line in the x direction

• The slope of the tangent line in the y direction