Understanding Derivatives of Composite Functions

The video tutorial explains how to determine the derivative of composite functions using the chain rule. It covers the derivatives of four functions: F(x), H(x), G(x), and J(x). Each function is broken down into its inner and outer components, and the chain rule is applied to find the derivative. The tutorial emphasizes the importance of identifying the inner function and its derivative, U Prime, to correctly apply the chain rule. The video also demonstrates the use of the power rule in conjunction with the chain rule to simplify the derivative expressions.

What is the first step in finding the derivative of a composite function?

• Apply the product rule

• Apply the quotient rule

• Identify the inner function

• Identify the outer function

In the function F(x) = -3 * cos(4x), what is the inner function?

• cos(4x)

• -3

• 4x

• cos(x)

What is the derivative of cos(x)?

• -cos(x)

• -sin(x)

• cos(x)

• sin(x)

In the function H(x) = -3 * cos(x^4), what is the derivative of the inner function x^4?

• 4x^3

• 3x^4

• x^3

• 4x

What is the derivative of sin(x)?

• -cos(x)

• cos(x)

• sin(x)

• -sin(x)

In the function G(x) = 4 * sin^3(x), what is the outer function?

• 4

• sin^3(x)

• 4 * sin(x)

• sin(x)

What is the derivative of sin^3(x) with respect to x?

• 3 * sin^2(x) * cos(x)

• 3 * cos(x)

• sin^2(x) * cos(x)

• 3 * sin(x) * cos(x)

In the function J(x) = 4 * sin(x^3), what is the derivative of the inner function x^3?

• 3x

• 3x^2

• x^2

• 3

What is the derivative of 4 * sin(x^3) with respect to x?

• 4 * cos(x^3) * 3x^2

• 4 * sin(x^3) * 3x^2

• 4 * cos(x^3) * x^2

• 4 * sin(x^3) * x^2

What rule is primarily used to find the derivative of composite functions?

• Product rule

• Power rule

• Quotient rule

• Chain rule