Partial Derivatives and Differentiation Rules

The video tutorial explains how to find first and second order partial derivatives of a function f(x, y). It begins with calculating the first order partial derivatives with respect to x and y, treating the other variable as a constant. The tutorial then proceeds to find the second order partial derivatives by differentiating the first order partials. The product rule and chain rule are applied during the process. The video concludes with a summary of the steps involved in finding partial derivatives.

What is the first step in finding second order partial derivatives?

• Find the second order partials directly.

• Find the first order partials.

• Integrate the function.

• Simplify the function.

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

• x

• y

• cosine y to the ninth

• sine y to the ninth

What is the derivative of 4x^5 with respect to x?

• 20x^5

• 5x^4

• 20x^4

• 4x^4

Which rule is applied when differentiating cosine y to the ninth with respect to y?

• Quotient rule

• Product rule

• Chain rule

• Power rule

What is the first order partial derivative of f with respect to y?

• -9y^8 cosine y to the ninth

• 20x^4 cosine y to the ninth

• -36x^5 y^8 sine y to the ninth

• 9y^8 sine y to the ninth

What is the result of differentiating the first order partial with respect to x with respect to y?

• -180x^4 y^8 sine y to the ninth

• 180x^5 y^9 sine y to the ninth

• 180x^4 y^8 sine y to the ninth

• -180x^5 y^9 sine y to the ninth

What rule is used to find the second order partial with respect to y?

• Product rule

• Chain rule

• Quotient rule

• Power rule

What is the derivative of y^8 with respect to y?

• 8y^7

• 8y^8

• 9y^8

• 7y^6

What is the coefficient of x^5 y^16 cosine y to the ninth in the second order partial with respect to y?

• -288

• -324

• 324

• 288

What is the simplified form of the second order partial derivative with respect to y?

• 324x^5 y^16 cosine y to the ninth + 288x^5 y^7 sine y to the ninth

• -324x^5 y^16 cosine y to the ninth - 288x^5 y^7 sine y to the ninth

• 324x^5 y^16 sine y to the ninth - 288x^5 y^7 cosine y to the ninth

• -324x^5 y^16 sine y to the ninth + 288x^5 y^7 cosine y to the ninth