The video tutorial explains how to determine the derivative of functions using the product rule. It starts with deriving f(x) = cosecant x times cotangent x, applying the product rule, and simplifying the result. The tutorial then moves on to g(x) = -3 cosecant x tangent x, deriving and simplifying g'(x). The video concludes with factoring techniques and helpful tips for understanding derivatives.

What is the initial function given for differentiation in the video?

• f(x) = cosecant x * cotangent x

• f(x) = sine x * cosine x

• f(x) = cosecant x * sine x

• f(x) = tangent x * secant x

Which rule is applied to differentiate the product of two functions in the video?

• Chain Rule

• Quotient Rule

• Product Rule

• Power Rule

What is the derivative of cotangent x as mentioned in the video?

• cosecant squared x

• -cosecant squared x

• secant squared x

• -secant squared x

What is the derivative of cosecant x as mentioned in the video?

• -secant x tangent x

• cosecant x cotangent x

• -cosecant x cotangent x

• secant x tangent x

What common factor is factored out from the derivative of f(x)?

• sine x

• tangent x

• cosine x

• cosecant x

What is the new function introduced for differentiation after f(x)?

• g(x) = 3 secant x * tangent x

• g(x) = -3 secant x * cotangent x

• g(x) = -3 cosecant x * tangent x

• g(x) = 3 cosecant x * cotangent x

What is the simplified form of the derivative of f(x) after factoring?

• negative cosecant x times (cosecant squared x + cotangent squared x)

• negative sine x times (sine squared x + cosine squared x)

• negative secant x times (secant squared x + tangent squared x)

• negative tangent x times (tangent squared x + secant squared x)

What is the derivative of tangent x as used in the video?

• -cosecant squared x

• -secant squared x

• cosecant squared x

• secant squared x

What is the greatest common factor factored out from the derivative of g(x)?

• 3 secant x

• 3 tangent x

• 3 sine x

• 3 cosecant x

What is the final expression for g'(x) after simplification and factoring?

• 3 secant x times (secant squared x + tangent squared x)

• 3 cosecant x times (cosecant squared x + cotangent squared x)

• 3 tangent x times (tangent squared x + secant squared x)

• 3 sine x times (sine squared x + cosine squared x)