Partial Derivatives and Functions

The video tutorial explains how to find the partial derivatives of a function f(x, y) = sin(x + y) where x and y are functions of s and t. The process involves using the chain rule to express the partial derivatives with respect to s and t in terms of s and t only. The tutorial demonstrates the step-by-step calculation and simplification of these derivatives, including factoring common terms for a more simplified expression.

What is the function f(x, y) given in the problem?

• f(x, y) = x + y

• f(x, y) = sin(x + y)

• f(x, y) = cos(x + y)

• f(x, y) = tan(x + y)

Which rule is applied to find the partial derivatives of f with respect to s and t?

• Quotient Rule

• Power Rule

• Product Rule

• Chain Rule

What is the partial derivative of f with respect to x when y is treated as a constant?

• x + y

• tan(x + y)

• cos(x + y)

• sin(x + y)

What is the expression for x in terms of s and t?

• x = s + t

• x = 5s - 6t

• x = s^5 t^6

• x = s^5 + t^6

What common factor can be factored out from the partial derivative of f with respect to s?

• 5 (s^5 t^6 + 5s - 6t)

• 5 tan(s^5 t^6 + 5s - 6t)

• 5 cos(s^5 t^6 + 5s - 6t)

• 5 sin(s^5 t^6 + 5s - 6t)

What is the partial derivative of f with respect to y when x is treated as a constant?

• x + y

• tan(x + y)

• cos(x + y)

• sin(x + y)

What is the expression for y in terms of s and t?

• y = s^5 t^6

• y = s^5 + t^6

• y = 5s - 6t

• y = s + t

What common factor can be factored out from the partial derivative of f with respect to t?

• 6 sin(s^5 t^6 + 5s - 6t)

• 6 cos(s^5 t^6 + 5s - 6t)

• 6 tan(s^5 t^6 + 5s - 6t)

• 6 (s^5 t^6 + 5s - 6t)

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

• 5 s^4 t^6

• 6 s^4 t^5

• 6 s^5 t^5

• 5 s^5 t^6

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

• -5

• 6

• -6

• 5