Understanding the Derivative of the Square Root of x

The video tutorial demonstrates the proof of the derivative of the square root of x. It begins by explaining the definition of a derivative and then simplifies the expression using conjugates. The tutorial concludes by applying the limit to derive the final result, showing that the derivative of x to the 1/2 power is 1/2x to the negative 1/2, consistent with the general derivative rule.

What is the derivative of the square root of x according to the definition of a derivative?

• The limit as delta x approaches infinity

• The limit as delta x approaches zero

• The limit as delta x approaches negative infinity

• The limit as delta x approaches one

In the limit definition of the derivative, what does f(x + delta x) represent?

• The function value at x plus a small change

• The derivative of the function

• The function value at x

• The original function

Why do we multiply by the conjugate in the simplification process?

• To eliminate the square root

• To add more terms

• To change the function

• To make the expression more complex

What happens to the terms x and -x in the simplification process?

• They are added together

• They remain unchanged

• They are multiplied

• They cancel each other out

What mathematical operation is used to simplify the expression after multiplying by the conjugate?

• Addition

• Multiplication

• Subtraction

• Division

What is the final expression for the derivative of the square root of x?

• 1 over square root of x

• 1 over 2 square root of x

• 1 over 2x

• 1 over x

What is the general property of derivatives that is consistent with the result of this proof?

• The derivative of x to the n is nx to the n plus 1

• The derivative of x to the n is nx to the n minus 1

• The derivative of x to the n is n plus x to the n

• The derivative of x to the n is n minus x to the n

What is the purpose of taking the limit as delta x approaches zero?

• To find the instantaneous rate of change

• To find the maximum value

• To find the minimum value

• To find the average rate of change

What is the result when delta x is set to zero in the final expression?

• The expression simplifies to a constant

• The expression becomes undefined

• The expression simplifies to zero

• The expression simplifies to 1 over 2 square root of x

What is the significance of proving the derivative for x to the 1/2 power?

• It proves the derivative is always positive

• It shows the derivative is always zero

• It confirms the derivative rule for all powers

• It demonstrates the derivative rule for fractional powers