Partial Derivatives and Expressions

The video tutorial explains how to calculate the partial derivatives of a function z with respect to variables s and t. It begins by introducing the problem and the need to find partial derivatives of x and y with respect to s and t. The tutorial then demonstrates the use of the chain rule to determine the partial derivatives of z with respect to s and t, providing step-by-step calculations and substitutions to express the derivatives in terms of s and t.

What are the expressions for x and y in terms of s and t?

• x = 7s + 9t, y = 2s + 5t

• x = 7s - 9t, y = 5s - 2t

• x = 9s - 7t, y = -2s + 5t

• x = 9s + 7t, y = -2s - 5t

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

• -7

• 7

• 5

• 9

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

• 9

• -2

• -7

• 5

Which rule is applied to find the partial derivative of z with respect to s?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

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

• 18y

• -2s

• -6x

• 9s

What is the result of substituting x and y into the expression for the partial derivative of z with respect to s?

• 486s - 198t

• -486s + 198t

• 198s - 486t

• -198s + 486t

What is the partial derivative of z with respect to t?

• 198s + 156t

• -198s - 156t

• -156s - 198t

• 156s + 198t

What is the expression for the partial derivative of z with respect to y?

• 9s

• 18y

• -6x

• -2s

What is the result of combining like terms for the partial derivative of z with respect to t?

• 156s + 198t

• -198s - 156t

• -156s - 198t

• 198s + 156t

What is the final step in the tutorial?

• Recording the results

• Applying the product rule

• Finding the derivative of x

• Differentiating with respect to y