Understanding Derivatives and Rational Exponents

The video tutorial explains how to find the derivative of a function involving square roots. It begins by rewriting square roots as rational exponents and simplifying the function. The power rule of differentiation is applied to find the derivative, which is then simplified. The derivative is rewritten using square roots, and the value of the derivative at x=1 is calculated. Finally, the result is verified graphically.

What is the first step in finding the derivative of the given function?

• Find the derivative at x = 1.

• Graph the function.

• Apply the power rule directly.

• Rewrite the function using rational exponents.

How do you express the square root of x using rational exponents?

• x^(1/2)

• x^2

• x^1

• x^(2/3)

What is the purpose of rewriting terms with negative exponents?

• To eliminate the need for exponents.

• To simplify the function for integration.

• To make the function more complex.

• To apply the power rule of differentiation.

What is the derivative of x^(3/2) using the power rule?

• 3/2 x^(1/2)

• x^(3/2)

• 2x^(3/2)

• 3x^(1/2)

How is the term x^(-5/2) rewritten using positive exponents?

• 5/x^(2)

• x^(5/2)

• 1/x^(5/2)

• x^(-5/2)

What is the simplified form of the derivative function using square roots?

• 3x^(1/2) + 10/x^(7/2)

• 3x^(1/2) - 10x^(7/2)

• 3√x - 10/x^(7/2)

• 3√x + 10x^(7/2)

What is the value of F'(1) for the given function?

• 1

• 7

• 0

• -7

What does F'(1) represent in the context of the function?

• The y-intercept of the function.

• The slope of the tangent line at x = 1.

• The maximum value of the function.

• The area under the curve.

How is the slope of the tangent line verified graphically?

• By observing the tangent line on the graph.

• By checking the intersection points.

• By plotting the derivative function.

• By calculating the area under the curve.

What is the significance of using rational exponents in differentiation?

• It simplifies the application of the power rule.

• It makes the function more complex.

• It changes the function's domain.

• It eliminates the need for differentiation.