Difference Quotient and Simplification Techniques

The video tutorial explains the concept of the difference quotient, a fundamental tool in calculus for finding the derivative of a function. It provides step-by-step examples for various functions, including linear, quadratic, and rational functions, as well as functions involving square roots. The tutorial emphasizes the process of substituting x with x + h, simplifying expressions, and using algebraic techniques like factoring and conjugates to find the difference quotient.

What is the difference quotient of the function f(x) = 7x?

• 7

• x

• h

• 0

For the function f(x) = 5x + 4, what is the first step in finding the difference quotient?

• Replace x with x + h

• Subtract 4

• Multiply by h

• Divide by h

After substituting x with x + h in f(x) = 5x + 4, what expression do you get?

• 5x + 4

• 5(x + h) + 4

• 5h + 4

• 5x + h + 4

What is the expanded form of (x + h)^2 when finding the difference quotient of f(x) = x^2?

• x^2 + h^2

• x^2 + 2xh + h^2

• 2x + h

• x^2 + 2x

What is the difference quotient of f(x) = x^2?

• 2x + h

• x^2

• h

• 2x

When simplifying the difference quotient of √x, what technique is used?

• Substitution

• Completing the square

• Using the conjugate

• Factoring

What is the difference quotient of f(x) = √x?

• 1/(√x + h)

• 1/(√x + √h)

• 1/(√x + h + √x)

• 1/(√x + h + √x)

For f(x) = 1/x, what is the common denominator used to simplify the difference quotient?

• x(x + h)

• x + h

• h

• x

What is the difference quotient of f(x) = 1/x?

• -1/(x(x + h))

• -1/(x^2)

• 1/(x + h)

• 1/x

In the expression 3x^2 + 4x - 5, what is the difference quotient?

• 6x + 3h + 4

• 3x^2 + 4x

• 6x + 4 + 3h

• 3x^2 + 4