Understanding Derivatives Using the Limit Definition

This video tutorial explains how to find the derivative of various functions using the definition of the derivative formula. It covers the limit process and provides examples for linear, quadratic, reciprocal, square root, and polynomial functions. The tutorial also addresses complex fractions and radicals, demonstrating step-by-step calculations to simplify expressions and find derivatives.

What is the limit definition of the derivative?

• The limit as x approaches 0 of f(x) divided by x

• The limit as x approaches infinity of f(x) divided by x

• The limit as h approaches 0 of f(x+h) minus f(x) divided by h

• The limit as h approaches infinity of f(x+h) minus f(x) divided by h

What is the derivative of the linear function f(x) = 5x - 4?

• 1

• 0

• 4

• 5

How do you find f(x+h) for f(x) = x^2?

• Replace x with h

• Replace x with x+h

• Add h to the function

• Multiply the function by h

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

• 2

• 2x

• x^2

• x

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

• Multiply by the conjugate

• Find the common denominator

• Subtract h from the function

• Add h to the function

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

• -1/x^2

• 1/x^2

• -x^2

• x^2

How do you simplify expressions with radicals in the numerator?

• Add the radicals

• Subtract the radicals

• Divide by the conjugate

• Multiply by the conjugate

What is the derivative of f(x) = √x?

• 1/√x

• √x/2

• 2√x

• 1/2√x

What is the first step in finding the derivative of a polynomial function using the limit definition?

• Add h to the function

• Replace x with x+h

• Multiply by the conjugate

• Find the common denominator

What is the derivative of f(x) = x^2 - 5x + 9?

• 2x + 5

• 2x - 5

• x + 5

• x - 5