Understanding Derivatives Using the Power Rule

The video tutorial explains how to determine the derivative of functions using the power rule. It begins by introducing the power rule and the need to rewrite radicals in rational exponent form. Two examples are provided: one involving a square root function and another with a fifth root function. Each example demonstrates the application of the power rule, simplification of negative exponents, and conversion back to radical form. The video concludes with a brief summary and a preview of the next lesson.

What is the Power Rule used for in calculus?

• Finding the integral of a function

• Determining the derivative of a function

• Calculating the limit of a function

• Solving differential equations

How do you express the cube root of x squared in rational exponent form?

• x to the power of 2/3

• x to the power of 3

• x to the power of 1/3

• x to the power of 3/2

What is the first step in finding the derivative of 9 times the square root of x?

• Differentiate with respect to y

• Use the chain rule

• Rewrite the function in rational exponent form

• Apply the Power Rule directly

After applying the Power Rule to 9x^(1/2), what is the next step?

• Multiply by the original function

• Convert the negative exponent to a positive one

• Leave the derivative as is

• Add a constant to the derivative

What is the final form of the derivative of 9 times the square root of x?

• 9/2 times the square root of x

• 9 times x to the power of -1/2

• 9/2 times x to the power of 1/2

• 9 times the square root of x

How is the function 15x^(2/5) initially rewritten?

• In rational exponent form

• As a square root

• As a logarithmic function

• Using the chain rule

What is the derivative of 15x^(2/5) after applying the Power Rule?

• 15x^(3/5)

• 15x^(-3/5)

• 6x^(3/5)

• 6x^(-3/5)

What is the final form of the derivative of 15x^(2/5) in radical form?

• 15 times the 5th root of x squared

• 6 times the 5th root of x cubed

• 15 over the 5th root of x squared

• 6 over the 5th root of x cubed

What does the denominator represent in the rational exponent form?

• The coefficient of the term

• The index of the root

• The base of the exponent

• The numerator of the exponent

What is the purpose of converting a negative exponent to a positive one?

• To simplify the expression

• To make the function continuous

• To apply the chain rule

• To find the integral