Exploring the Chain Rule for Composite Functions

Professor Dave introduces the chain rule, a crucial concept in calculus for differentiating composite functions. He explains how to apply the chain rule through various examples, including trigonometric and polynomial functions. The tutorial covers the importance of differentiating the outer function first and then multiplying by the derivative of the inner function. Advanced examples demonstrate combining the chain rule with the product and quotient rules. The video emphasizes the importance of order in applying the chain rule and provides practice problems to reinforce understanding.

What is the chain rule used for?

• To solve algebraic equations

• To integrate functions

• To differentiate composite functions

• To differentiate simple polynomials

In the function sqrt(x^2 + 1), what is the inner function?

• x^2 + 1

• x^2

• sqrt(x^2 + 1)

• 1

When applying the chain rule, what do you do after differentiating the outer function?

• Multiply by the derivative of the inner function

• Differentiate the inner function

• Integrate the outer function

• Simplify the expression

What is the derivative of sin(x^2) using the chain rule?

• 2x cos(x^2)

• sin(x^2)

• cos(x^2)

• 2x sin(x^2)

How do you handle a function like (x^4 - 1)^50 using the chain rule?

• Use the power rule only

• Differentiate directly

• Use the quotient rule

• Apply the chain rule and then the power rule

What additional rule is needed to differentiate (x-1)/(x+1)^2?

• Product rule

• Quotient rule

• Power rule

• Sum rule

In the expression (x+3)^2 * (x^2-4)^3, which rule is applied first?

• Power rule

• Product rule

• Quotient rule

• Chain rule

What is the derivative of (x^2 - 4)^3 using the chain rule?

• 3(x^2 - 4)^3

• 6x(x^2 - 4)^3

• 6x(x^2 - 4)^2

• 3(x^2 - 4)^2

How do you differentiate sin(cos(tan(x)))?

• Use the product rule

• Apply the chain rule three times

• Apply the chain rule twice

• Apply the chain rule once

What is the final derivative of sin(cos(tan(x)))?

• sin(cos(tan(x))) * sin(tan(x)) * sec^2(x)

• sin(cos(tan(x))) * cos(tan(x)) * sec^2(x)

• cos(cos(tan(x))) * cos(tan(x)) * sec^2(x)

• cos(cos(tan(x))) * sin(tan(x)) * sec^2(x)