Partial Derivatives and Their Interpretation

The video tutorial explains the concept of partial derivatives for a function of two variables, focusing on calculating derivatives with respect to X and Y. It uses the chain rule and trigonometric functions to derive the expressions. The tutorial also discusses the geometric interpretation of partial derivatives as slopes of tangent lines in the X and Y directions. Graphical representations are used to illustrate how these slopes change at different points on the surface.

What is the function given in the video?

• f(x, y) = 7 * e^(3x^2 - 5) * sin(4y)

• f(x, y) = 7 * e^(3x^2 + 5) * sin(4y)

• f(x, y) = 7 * e^(3x^2 + 5) * cos(4y)

• f(x, y) = 7 * e^(3x^2 - 5) * cos(4y)

When finding the partial derivative with respect to x, what is treated as a constant?

• x and e^(3x^2 - 5)

• y and cos(4y)

• 7 and cos(4y)

• 7 and e^(3x^2 - 5)

What rule is applied to find the partial derivative with respect to x?

• Chain rule

• Power rule

• Product rule

• Quotient rule

What is the partial derivative of the function with respect to x?

• 28 e^(3x^2 - 5) sin(4y)

• 28x e^(3x^2 - 5) sin(4y)

• 42 e^(3x^2 - 5) cos(4y)

• 42x e^(3x^2 - 5) cos(4y)

What is considered constant when finding the partial derivative with respect to y?

• x and e^(3x^2 - 5)

• y and cos(4y)

• 7 and e^(3x^2 - 5)

• 7 and sin(4y)

What is the partial derivative of the function with respect to y?

• -42 e^(3x^2 - 5) sin(4y)

• -28 e^(3x^2 - 5) sin(4y)

• 28 e^(3x^2 - 5) cos(4y)

• 42 e^(3x^2 - 5) cos(4y)

What does the partial derivative with respect to x represent?

• Slope of the tangent line in the positive y direction

• Slope of the tangent line in the positive x direction

• Slope of the tangent line in the negative y direction

• Slope of the tangent line in the negative x direction

What does the partial derivative with respect to y represent?

• Slope of the tangent line in the positive y direction

• Slope of the tangent line in the positive x direction

• Slope of the tangent line in the negative y direction

• Slope of the tangent line in the negative x direction

In the graphical analysis, what does a positive slope indicate?

• The line moves downward

• The line moves sideways

• The line moves upward

• The line remains constant

What is the approximate value of the partial derivative with respect to y at a point where the tangent line has a negative slope?

• 0.6798

• 0.9818

• 0.9582

• 0.3762