Differentiation and Partial Derivatives

The video tutorial explains how to find partial derivatives of functions. It begins with an introduction to the concept of partial derivatives, followed by detailed steps to calculate them with respect to x and y using the chain rule. The tutorial includes two examples: a rational function and an exponential function, demonstrating the application of the chain rule in each case. The video concludes with a summary of the key points covered.

What is the first step in finding a partial derivative?

• Integrate the function

• Treat one variable as a constant and differentiate with respect to another

• Differentiate with respect to all variables simultaneously

• Treat all variables as constants

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

• Neither x nor y

• y

• Both x and y

• x

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

• Product Rule

• Quotient Rule

• Chain Rule

• Power Rule

In the function 2/(xy-4)^7, what is the inner function labeled as?

• u

• w

• v

• z

What is the result of differentiating xy with respect to y, treating x as a constant?

• 0

• y

• x

• 1

What is the inner function in the exponential function e^(7x-4y)?

• e

• 7x-4y

• 4y

• 7x

What is the derivative of 7x with respect to x?

• 7

• 1

• 0

• x

What is the final simplified form of the partial derivative of e^(7x-4y) with respect to x?

• 7e^(7x-4y)

• 0

• -4e^(7x-4y)

• e^(7x-4y)

When differentiating e^(7x-4y) with respect to y, what is the derivative of -4y?

• 1

• 0

• 4

• -4

What is the result of differentiating e^(7x-4y) with respect to y?

• 7e^(7x-4y)

• 0

• -4e^(7x-4y)

• e^(7x-4y)