Understanding Derivatives and the Chain Rule

The video tutorial explains how to find the derivative of a composite function using the chain rule. It begins by introducing the function f(x) = 2 * arctan(sin(2x)) and identifies it as a composite function. The tutorial then breaks down the process of finding the derivative by first identifying the inner function and its derivative. It uses the chain rule to calculate the derivative of the entire function and simplifies the expression. Finally, it provides guidance on how to enter the solution into an online homework system.

What is the main goal when analyzing the function f(x) in this tutorial?

• To graph f(x)

• To solve f(x) for x

• To find the derivative f'(x)

• To find the integral of f(x)

Which function is identified as the outer function in the composite function f(x)?

• Exponential function

• Sine function

• Inverse tangent function

• Cosine function

What is the inner function u in the composite function f(x)?

• exponential(2x)

• sine(2x)

• cosine(2x)

• tangent(2x)

What rule is used to find the derivative of the inner function u?

• Chain rule

• Quotient rule

• Product rule

• Power rule

What is the derivative of the inner function u, which is sine(2x)?

• sine(2x)

• 2 sine(2x)

• 2 cosine(2x)

• cosine(2x)

What is the expression for f'(x) after applying the derivative formula?

• 2 sine(2x) / (1 + cosine^2(2x))

• 2 cosine(2x) / (1 + sine^2(2x))

• 4 cosine(2x) / (1 + sine^2(2x))

• 4 sine(2x) / (1 + sine^2(2x))

What is the simplified form of the numerator in the expression for f'(x)?

• 2 sine(2x)

• 4 cosine(2x)

• 4 sine(2x)

• 2 cosine(2x)

How is sine^2(2x) represented in the final expression for f'(x)?

• cosine(2x)^2

• sine^2(2x)

• 2 sine(2x)

• sine(2x)^2

What is the recommended method for entering the derivative into an online homework system?

• Using a calculator

• Typing manually

• Using the math palette

• Copying from a textbook

What is the denominator in the final expression for f'(x)?

• 1 + tangent^2(2x)

• 1 + sine^2(2x)

• 1 + exponential(2x)

• 1 + cosine^2(2x)