Chain Rule in Differentiation

This video tutorial introduces the chain rule of differentiation, a technique for finding derivatives of composite functions. It explains the chain rule's application, using examples to demonstrate how to differentiate composite functions by identifying inner and outer functions. The tutorial also covers the power rule in conjunction with the chain rule and provides advanced examples to solidify understanding. Key concepts include the differentiation of composite functions, the use of Leibniz notation, and the application of the power rule.

What is the primary purpose of the chain rule in differentiation?

• To solve differential equations

• To integrate composite functions

• To find derivatives of composite functions

• To differentiate simple functions

In the chain rule, what does the derivative of the outer function evaluated at the inner function represent?

• The derivative of the inner function

• The integral of the outer function

• The original function

• The derivative of the outer function

How is the chain rule expressed using Leibniz notation?

• dy/dx = du/dy * dx/du

• dy/dx = dx/dy * du/dx

• dy/dx = dy/du * du/dx

• dy/dx = dy/d * dx

In the example where f(x) = (2x + 1)^3, what is the inner function?

• 2x + 1

• 2x

• x^3

• 3x^2

What is the derivative of the inner function U when U = 4x^3 + 5?

• 8x

• 4x^2

• 3x^2

• 12x^2

When applying the chain rule to a function with a rational exponent, what is the first step?

• Differentiate the inner function

• Identify the outer function

• Differentiate the outer function

• Identify the inner function

In the example with the function H(x) = 11e^(3x^8), what is the derivative of the inner function U?

• 24x^7

• 3x^8

• 11e^x

• 8x^3

What is the derivative of G(x) = 4 * ln(x^3 - 4x^2 + 5) with respect to x?

• 4/(x^3 - 4x^2 + 5) * (3x^2 - 8x)

• 4 * ln(x^3 - 4x^2 + 5)

• 4x^2 - 8x + 5

• 3x^2 - 8x

Which of the following functions can be differentiated using the chain rule?

• f(x) = x^2 + 3x + 5

• f(x) = sin(x^2)

• f(x) = 3x + 7

• f(x) = ln(x)

What is the result of differentiating the function H(x) = 11e^(3x^8) using the chain rule?

• 24x^7e^(3x^8)

• 3x^8e^(11x)

• 264x^7e^(3x^8)

• 11e^(3x^8)