Understanding Derivatives Using the Limit Definition

What is the derivative of the function f(x) = 3x - 2 using the limit definition?

• 3

• 1

• 2

• 0

When using the limit definition, what happens to the terms 3x and -3x in the derivative calculation of 3x - 2?

• They multiply to 9x^2

• They remain unchanged

• They add up to 6x

• They cancel each other out

What is the derivative of f(x) = √x using the limit definition?

• √x/2

• 1/(2√x)

• 2√x

• 1/√x

Why is the conjugate used in the limit definition for the derivative of √x?

• To simplify the expression

• To eliminate the square root

• To find a common denominator

• To multiply the terms

What is the derivative of f(x) = 1/x using the limit definition?

• -x^2

• x^2

• 1/x^2

• -1/x^2

In the derivative calculation of 1/x, what happens to the terms x and -x?

• They multiply to x^2

• They add up to 2x

• They cancel each other out

• They remain unchanged

What is the derivative of f(x) = x^2 + x + 1 using the limit definition?

• 2x + 1

• 2x + 2

• x + 1

• x^2 + 1

In the derivative calculation of x^2 + x + 1, what happens to the terms x^2 and -x^2?

• They cancel each other out

• They add up to 2x^2

• They multiply to x^4

• They remain unchanged

What is the purpose of factoring out H in the limit definition for the derivative of x^2 + x + 1?

• To find a common denominator

• To multiply the terms

• To eliminate the variable

• To simplify the expression

What happens to the term involving H as H approaches 0 in the derivative calculation of x^2 + x + 1?

• It becomes infinite

• It disappears

• It remains unchanged

• It doubles