Understanding Derivatives and Tangent Lines

The video tutorial explains how to find the derivative of the function F(x) = 3x * inverse sin(x) using the product rule. It then evaluates the derivative at x = 0.4 to find the slope of the tangent line at that point. The process involves applying the product rule, calculating the derivative, and using a calculator for numerical approximation. Finally, the result is verified graphically by plotting the function and its tangent line.

What is the initial function given in the video?

• F(x) = 3x - inverse sin(x)

• F(x) = 3x + inverse sin(x)

• F(x) = 3x * inverse sin(x)

• F(x) = 3x * sin(x)

Which rule is applied to find the derivative of the given function?

• Product Rule

• Power Rule

• Chain Rule

• Quotient Rule

What is the first function (f) in the product rule application?

• x^2

• 3x

• inverse sin(x)

• 3x^2

What is the derivative of 3x with respect to x?

• 3

• 1

• x

• 0

What is the derivative of inverse sin(x) with respect to x?

• 1 / (1 + x^2)

• x / sqrt(1 - x^2)

• 1 / sqrt(1 - x^2)

• x / (1 + x^2)

What is the value of F'(0.4) approximately?

• 3.54

• 0.54

• 1.54

• 2.54

In which mode should the calculator be set to find the inverse sin of 0.4?

• Decimal

• Gradian

• Degree

• Radian

What does the derivative value at x = 0.4 represent graphically?

• The y-intercept of the tangent line

• The slope of the tangent line

• The x-intercept of the tangent line

• The area under the curve

What color is the original function graphed in?

• Yellow

• Blue

• Red

• Green

How is the slope of the tangent line verified graphically?

• By calculating the area under the tangent line

• By finding another point on the tangent line

• By measuring the angle

• By checking the y-intercept