Derivatives and Their Simplifications

The video tutorial explains how to find the second and third derivatives of a given function. It begins with calculating the first derivative using the power rule, then proceeds to the second derivative, simplifying terms and discussing common denominators. Finally, it covers the third derivative, again simplifying and factoring terms. The tutorial emphasizes the use of the power rule and touches on the chain rule, providing equivalent expressions for the derivatives.

What is the first step in finding the second and third derivatives of a function?

• Start by finding the first derivative.

• Use the chain rule immediately.

• Find the second derivative directly.

• Rewrite the function in terms of y.

How do you rewrite the term x raised to the -3rd power to apply the power rule?

• As x raised to the -5th power.

• As x raised to the 5th power.

• As x raised to the -3rd power.

• As x raised to the 3rd power.

What is the first derivative of the function given in the video?

• 20x to the 4th + 3x to the -4th

• 20x to the 3rd + 3x to the -5th

• 20x to the 5th + 3x to the -3rd

• 20x to the 4th - 3x to the -4th

What is the second derivative of the function after simplification?

• 80x to the 3rd - 12/x to the 5th

• 80x to the 4th - 12/x to the 6th

• 80x to the 3rd + 12/x to the 5th

• 80x to the 2nd - 12/x to the 4th

What is the common denominator used to combine terms in the second derivative?

• x to the 3rd

• x to the 4th

• x to the 5th

• x to the 6th

What is the third derivative of the function before simplification?

• 240x squared + 60/x to the 6th

• 240x to the 3rd + 60/x to the 5th

• 240x squared - 60/x to the 6th

• 240x to the 4th + 60/x to the 7th

What is the common factor in the numerator of the third derivative?

• 20

• 40

• 80

• 60

What is the final form of the third derivative after factoring?

• 6x to the 8th + 2/x to the 7th

• 3x to the 8th + 1/x to the 5th

• 5x to the 8th + 2/x to the 6th

• 4x to the 8th + 1/x to the 6th

Why might the last two steps of simplification not be necessary?

• They are incorrect.

• They are just alternative ways to express the function.

• They are not part of the original problem.

• They make the function more complex.

What rule is primarily used to find the derivatives in this tutorial?

• Quotient rule

• Product rule

• Power rule

• Chain rule