Understanding Derivatives Using Limit Definition

The video tutorial explains how to find the derivative of the function f(x) = 3/x at x = 4 using the limit definition of a derivative. It outlines two methods: finding the derivative function first and then substituting x = 4, or directly substituting x = 4 into the limit definition. The tutorial walks through the process of substituting values, finding a common denominator, simplifying the expression, and calculating the limit as h approaches zero. Finally, it verifies the result graphically, showing that the slope of the tangent line at x = 4 is -3/16.

What is the main goal of the problem discussed in the video?

• To solve the equation f(x) = 3/x for x = 4

• To find the derivative of f(x) = 3/x at x = 4 using the limit definition

• To graph the function f(x) = 3/x

• To find the integral of f(x) = 3/x at x = 4

Which method is used to find the derivative in the video?

• Chain Rule

• Limit Definition

• Product Rule

• Power Rule

What expression is set up to find f'(4) using the limit definition?

• f'(4) = lim (h -> 0) [(f(4+h) - f(4))/h]

• f'(4) = 3/4

• f'(4) = 3/x

• f'(4) = 3/(4+h)

What is the least common denominator used to combine the fractions in the expression?

• h

• 4(4 + h)

• 4 + h

• 4

What is the result of distributing the negative three in the expression?

• -12 - 3h

• 12 + 3h

• 12 - 3h

• -12 + 3h

What happens to the h terms in the numerator and denominator during simplification?

• They are added together

• They are multiplied

• They remain unchanged

• They cancel each other out

What is the final value of the derivative f'(4)?

• -3/4

• -3/16

• 3/16

• 3/4

What does the derivative value represent in the context of the graph?

• The area under the curve

• The x-intercept of the graph

• The slope of the tangent line at x = 4

• The y-intercept of the graph

How is the derivative value verified in the video?

• By solving an equation

• By using a calculator

• Graphically, by checking the slope of the tangent line

• By comparing with a table of values

What is the point on the graph of f(x) = 3/x at x = 4?

• (4, 1/3)

• (4, 1)

• (4, 3)

• (4, 3/4)