Understanding Derivatives Using the Power Rule

The video tutorial explains how to determine the derivative of a function using the power rule. It begins by introducing the power rule and then demonstrates how to rewrite a complex function to apply the rule effectively. The tutorial walks through finding the derivative of each term, simplifying the resulting expression, and converting it back to a more familiar form. The process includes handling rational exponents and simplifying cube roots, providing a comprehensive review of derivative calculation techniques.

What does the Power Rule state about the derivative of x raised to the power of n?

• It is equal to n times x raised to the power of n.

• It is equal to n times x raised to the power of n minus 1.

• It is equal to n times x raised to the power of n plus 1.

• It is equal to x raised to the power of n minus 1.

Why do we rewrite the function before applying the Power Rule?

• To make the function more complex.

• To simplify the function for easier differentiation.

• To change the function's domain.

• To eliminate any constants in the function.

How is the term 4/x^(1/3) rewritten to apply the Power Rule?

• 4x^(1)

• 4x^(-1/3)

• 4x^(1/3)

• 4x^(3)

What is the derivative of a constant term?

• Zero.

• One.

• The constant itself.

• The constant times x.

What is the derivative of 7x^2 using the Power Rule?

• 14x^2

• 7x

• 7x^3

• 14x

What is the simplified form of the derivative of 1/5x^(3/2)?

• 3/10 x^(1/2)

• 1/5 x^(1/2)

• 3/10 x^(3/2)

• 1/5 x^(3/2)

How is x^(1/2) expressed in radical form?

• Cube root of x

• Square root of x

• Fourth root of x

• x squared

What is the final simplified form of the derivative of the function?

• 3 square root x/10 - 4/3 x^(4/3) - 14x

• 3 square root x/10 - 4/3 x^(1/3) + 14x

• 3 square root x/10 - 4/3 x^(4/3) + 14x

• 3 square root x/10 + 4/3 x^(4/3) + 14x

What is the cube root of x^4 simplified to?

• x cube root x

• x^2

• x^3

• x^4

Why is it important to simplify expressions involving radicals?

• To make them more complex.

• To make them easier to understand and work with.

• To change their meaning.

• To increase their value.