Chain Rule and Derivatives

The video tutorial explains how to determine the derivative of various functions using the chain rule. It covers the process of identifying inner functions and applying the chain rule to composite functions. The tutorial provides step-by-step examples for functions F(x), H(x), G(x), and J(x), demonstrating how to rewrite functions, find derivatives, and simplify results.

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

• Differentiate directly

• Identify the inner function

• Use the product rule

• Identify the outer function

Tags

CCSS.HSF-BF.A.1C

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

• cos(x)

• cos^4(x)

• -3

• x

What is the derivative of cos(x) with respect to x?

• cos(x)

• -cos(x)

• -sin(x)

• sin(x)

When applying the chain rule, what do you multiply the derivative of the outer function by?

• The derivative of the inner function

• The inner function

• The original function

• The derivative of the outer function

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

• x^3

• 3x^2

• 4x

• 4x^3

Tags

CCSS.HSF-BF.B.4A

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

• sin(x)

• -sin(x)

• -cos(x)

• cos(x)

Tags

CCSS.HSF-BF.B.4A

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

• sin(x)

• sin^3(x)

• x

• 4

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

• x^3

• x^2

• 3x^2

• 3x

What is the result of differentiating 4 * sin(x^3) using the chain rule?

• 12x^3 * cos(x)

• 4x^2 * sin(x^3)

• 12x^2 * cos(x^3)

• 4x^3 * sin(x)

What is the final simplified form of the derivative of J(x) = 4 * sin(x^3)?

• 12x^3 * cos(x)

• 4x^2 * sin(x^3)

• 12x^2 * cos(x^3)

• 4x^3 * sin(x)