Understanding Derivatives and the Chain Rule

The video tutorial explains how to determine the second derivative of a function. It begins by rewriting a radical using a rational exponent and discusses the chain rule, clarifying when it is necessary to apply it. The tutorial then demonstrates how to find the first derivative and subsequently the second derivative, emphasizing the importance of simplifying the result. The explanation includes converting negative exponents to positive and rewriting the final answer in radical form, offering multiple equivalent forms of the solution.

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

• Apply the chain rule directly.

• Rewrite the radical using a rational exponent.

• Simplify the function using algebraic identities.

• Differentiate the function twice.

What is the rewritten form of x^(1/3) using a rational exponent?

• x^(1/3)

• x^(1/2)

• x^3

• x^(3/2)

When is it unnecessary to apply the chain rule?

• When the derivative of the inner function is one.

• When the function is a constant.

• When the derivative of the inner function is zero.

• When the function is a polynomial.

What is the purpose of finding the first derivative in this problem?

• To simplify the original function.

• To solve for the variable x.

• To apply the chain rule.

• To prepare for finding the second derivative.

What rule is used to find the first derivative of x^(1/3)?

• Chain rule

• Product rule

• Quotient rule

• Power rule

What is the exponent of x in the first derivative of x^(1/3)?

• -2/3

• -1/3

• 1/3

• 2/3

How do you handle a negative exponent when simplifying the second derivative?

• Convert it to a positive exponent by moving it to the denominator.

• Multiply the exponent by negative one.

• Leave it as is.

• Add one to the exponent.

What is the coefficient of x in the second derivative of x^(1/3)?

• 1/3

• -2/9

• 2/9

• -1/3

What is the final form of the second derivative if the original function was given in radical form?

• As a logarithmic function.

• In radical form.

• As a fraction with a negative exponent.

• As a polynomial.

Which form of the second derivative might be expected by an instructor?

• The form as a logarithmic function.

• The form with a negative exponent.

• The form as a polynomial.

• The form in radical notation.