Derivatives and Chain Rule Concepts

The video tutorial explains how to find the derivatives of composite functions using the chain rule. It covers the derivatives of inverse trigonometric functions, specifically inverse tangent and inverse secant. The tutorial provides step-by-step calculations, including simplification techniques, to derive f'(x) and g'(x). Key concepts such as the chain rule, derivative formulas, and simplification of expressions are discussed.

What is the inner function u when finding the derivative of f(x) = inverse tangent of 2x^5?

• 5x^2

• x^2

• x^5

• 2x^5

What is the derivative of the inner function u = 2x^5 with respect to x?

• 10x^4

• 2x^4

• 5x^4

• x^4

Tags

CCSS.HSA.APR.A.1

Which of the following is the correct expression for f'(x) after applying the chain rule?

• 10x^4 / (1 + 4x^10)

• 10x^4 / (1 + 2x^5)

• 10x^4 / (1 + x^10)

• 10x^4 / (1 + 4x^5)

Tags

CCSS.HSA.APR.A.1

Why can't we simplify 10x^4 and 4x^10 in the expression for f'(x)?

• Because they are not like terms

• Because they are separated by addition

• Because they are constants

• Because they are in different functions

What is the inner function u when finding the derivative of g(x) = inverse secant of 3x^3?

• x^2

• 3x^2

• x^3

• 3x^3

What is the derivative of the inner function u = 3x^3 with respect to x?

• x^2

• 6x^2

• 9x^2

• 3x^2

Which of the following is the correct expression for g'(x) after applying the chain rule?

• 9x^2 / (|3x^3| * sqrt(9x^6 - 1))

• 9x^2 / (|x^3| * sqrt(9x^6 - 1))

• 9x^2 / (|3x^3| * sqrt(x^6 - 1))

• 9x^2 / (|x^3| * sqrt(x^6 - 1))

Tags

CCSS.HSA.APR.D.6

How is the expression for g'(x) simplified in terms of x?

• 3 / (|3x| * sqrt(x^6 - 1))

• 3 / (|x| * sqrt(x^6 - 1))

• 3 / (|3x| * sqrt(9x^6 - 1))

• 3 / (|x| * sqrt(9x^6 - 1))

Tags

CCSS.6.EE.A.2C

What mathematical operation allows the simplification of 9x^2 and |3x^3| in g'(x)?

• Addition

• Multiplication

• Division

• Subtraction

What is the final simplified form of g'(x)?

• 9 / (|3x| * sqrt(9x^6 - 1))

• 3 / (|3x| * sqrt(9x^6 - 1))

• 9 / (|x| * sqrt(9x^6 - 1))

• 3 / (|x| * sqrt(9x^6 - 1))