Calculus: Derivatives and Quotient Rule

The video tutorial explains how to find the derivative of the function f(x) = 2tan(x)/x using the quotient rule. It demonstrates the calculation of the derivative at x = pi/4, simplifies the expression, and verifies the result graphically. The tutorial also covers trigonometric function values and provides both exact and approximate results.

What is the function given in the problem that requires finding its derivative?

• f(x) = 2cos(x)/x

• f(x) = 2tan(x)/x

• f(x) = 2cot(x)/x

• f(x) = 2sin(x)/x

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

• Product Rule

• Chain Rule

• Quotient Rule

• Power Rule

In the quotient rule, what is the expression for the derivative of a function f(x)/g(x)?

• g(x)f'(x) - f(x)g'(x) / g(x)^2

• f(x)g'(x) - g(x)f'(x) / f(x)^2

• f'(x)g(x) + f(x)g'(x) / g(x)^2

• g(x)f'(x) + f(x)g'(x) / f(x)^2

What is the derivative of 2tan(x) with respect to x?

• 2sec(x)

• 2sec^2(x)

• 2tan(x)sec(x)

• 2cos^2(x)

What is the value of tangent at π/4?

• 2

• 0

• 1

• √2

What is the exact value of f'(π/4) after simplification?

• 8π - 1/π^2

• 16π - 2/π^2

• 2π - 1/π^2

• 4π - 2/π^2

What is the approximate decimal value of f'(π/4) to three decimal places?

• 1.851

• 0.577

• 2.718

• 3.142

What is the significance of the slope of the tangent line at x = π/4?

• It represents the maximum value of the function.

• It indicates the rate of change of the function at that point.

• It shows the minimum value of the function.

• It is irrelevant to the function's behavior.

What is the graphical representation of the derivative at x = π/4?

• A tangent line with slope 1.851

• A horizontal line

• A vertical line

• A curve with no slope

What is the color of the graph representing the function in the video?

• Red

• Green

• Blue

• Yellow