Partial Derivatives and Chain Rule

This video tutorial introduces the chain rule for functions of multiple variables, focusing on the second case where variables depend on two parameters. It explains the concept of partial derivatives and demonstrates the calculation of these derivatives with respect to parameters t and s using a detailed example. The lesson concludes with a brief overview of the next topic, which involves generalizing the chain rule for functions of n variables depending on m parameters.

What is the main topic of the second theorem discussed in the video?

• Chain rule for functions of multiple variables

• Limits of single-variable functions

• Integration of multivariable functions

• Optimization of functions

In the second case of the chain rule, what do the variables x and y depend on?

• Three parameters

• Two parameters, t and s

• Only one parameter

• No parameters

What is the partial derivative of Z with respect to t in the given example?

• 2ts * e^x * sin(y) + s^2 * e^x * cos(y)

• e^x * sin(y) + e^x * cos(y)

• 2ts * e^x * cos(y) + s^2 * e^x * sin(y)

• e^x * sin(y) * cos(y)

What is the function Z in the example provided?

• Z = e^y * cos(x)

• Z = e^y * sin(x)

• Z = e^x * sin(y)

• Z = e^x * cos(y)

What is the partial derivative of Z with respect to x in the example?

• e^x * sin(y)

• sin(y)

• e^x * cos(y)

• cos(y)

What is the partial derivative of Z with respect to y in the example?

• cos(y)

• e^x * sin(y)

• e^x * cos(y)

• sin(y)

What is the value of x in terms of s and t in the example?

• x = s * t

• x = s + t^2

• x = s * t^2

• x = s^2 * t

What is the value of y in terms of s and t in the example?

• y = s * t^2

• y = s^2 * t

• y = s * t

• y = s + t^2

What is the partial derivative of Z with respect to s in the given example?

• e^(s*t^2) * sin(s^2 * t) + e^(s*t^2) * cos(s^2 * t)

• t^3 * e^(s*t^2) * sin(s^2 * t) + 2s * e^(s*t^2) * cos(s^2 * t)

• t^3 * e^(s*t^2) * cos(s^2 * t) + 2s * e^(s*t^2) * sin(s^2 * t)

• e^(s*t^2) * sin(s^2 * t) * cos(s^2 * t)

What will be discussed in the next lesson?

• Limits of single-variable functions

• Generalization of the chain rule for functions of n variables

• Integration of multivariable functions

• Optimization of functions