Understanding Derivatives Using the Quotient Rule

The video tutorial explains how to determine the derivative of a function using the quotient rule. It begins with an introduction to the problem and the application of the quotient rule, which involves differentiating the numerator and denominator separately. The tutorial then sets up the derivative function and simplifies the expression, focusing on the numerator. Finally, it presents the final derivative expression and demonstrates factoring out the greatest common factor. The tutorial aims to provide a clear understanding of the process of finding derivatives using the quotient rule.

What is the initial function given for differentiation?

• f(x) = 2x / (2 + 3 cot(x))

• f(x) = -2x / (2 - 3 cot(x))

• f(x) = -2x / (2 + 3 cot(x))

• f(x) = 2x / (2 - 3 cot(x))

Which rule is used to find the derivative of a quotient of two functions?

• Power Rule

• Product Rule

• Quotient Rule

• Chain Rule

In the quotient rule, what is the expression for the derivative of u/v?

• v * u' - u * v' / v^2

• u * v' - v * u' / u^2

• u' * v + u * v' / v^2

• v' * u - u' * v / u^2

What is the derivative of the numerator -2x in the given function?

• -1

• 0

• 2

• -2

What is the derivative of the constant term 2 in the denominator?

• 2

• 0

• 1

• -2

What is the derivative of cotangent x?

• -secant squared x

• -cosecant squared x

• secant squared x

• cosecant squared x

What is the result of distributing -2 in the expression -2(2 - 3 cot(x))?

• -4 + 6 cot(x)

• 4 - 6 cot(x)

• -4 - 6 cot(x)

• 4 + 6 cot(x)

What is the greatest common factor that can be factored out from the numerator?

• 3

• 6

• 2

• 1

What is the final form of the derivative function after factoring?

• 2(3x csc^2(x) - 3 cot(x) + 2) / (2 - 3 cot(x))^2

• 3(2x csc^2(x) - 2 cot(x) + 3) / (2 - 3 cot(x))^2

• 3(2x csc^2(x) + 2 cot(x) - 3) / (2 - 3 cot(x))^2

• 2(3x csc^2(x) + 3 cot(x) - 2) / (2 - 3 cot(x))^2

What is the denominator of the derivative function?

• (2 - 3 csc(x))^2

• (2 + 3 csc(x))^2

• (2 + 3 cot(x))^2

• (2 - 3 cot(x))^2