Composite Functions and Derivatives

The video tutorial explains how to find the derivative of a composite function involving cube roots. It begins by rewriting the function using rational exponents and identifies the need to apply the chain rule. The process involves identifying inner functions and calculating their derivatives. The tutorial then simplifies the derivative expression and presents it in both rational and radical forms, ensuring clarity in understanding the mathematical concepts involved.

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

• Differentiate directly

• Convert radicals to rational exponents

• Apply the product rule

• Use the quotient rule

What rule is applied when dealing with composite functions?

• Power rule

• Quotient rule

• Chain rule

• Product rule

In the context of the chain rule, what is the 'inner function' referred to as?

• U

• Z

• V

• W

How do you find the derivative of a composite function like sin(x^(1/3))?

• Apply the chain rule twice

• Differentiate directly

• Use the quotient rule

• Use the product rule

What is the derivative of x^(1/3) in terms of rational exponents?

• 1/3 * x^(-2/3)

• 3 * x^(2/3)

• 1/3 * x^(1/3)

• x^(1/3)

What is the purpose of writing the derivative using positive exponents?

• To avoid using the chain rule

• To apply the product rule

• To simplify the expression

• To make it easier to integrate

What is the final form of the derivative function expressed in rational exponents?

• cos(x^(1/3)) / (3x^(1/3) * s(x^(1/3))^(1/3))

• sin(x^(1/3)) / (9x^(2/3) * s(x^(1/3))^(2/3))

• cos(x^(1/3)) / (9x^(2/3) * s(x^(1/3))^(2/3))

• sin(x^(1/3)) / (3x^(1/3) * s(x^(1/3))^(1/3))

How can the derivative be expressed in radical form?

• As the cube root of x

• As the fourth root of x

• As the cube root of the square of x

• As the square root of the cube of x

What is the index of the radical when expressing x^(2/3) in radical form?

• 2

• 3

• 1

• 4

What happens to the exponent when moving a term to the denominator?

• It becomes positive

• It halves

• It doubles

• It becomes negative