Logarithmic Differentiation and Trigonometric Identities

The video tutorial explains how to find the derivative of a function using logarithmic differentiation. The function involves a cosine term raised to a power that is also a function of x. The process involves taking the natural log of both sides, applying the chain and product rules, and simplifying the expression using trigonometric identities. The final derivative is expressed in terms of x.

Why is logarithmic differentiation used in this problem?

• Because the exponent is a constant.

• Because the exponent is a function of x.

• Because the base is a constant.

• Because the base is a function of x.

What is the first step in applying logarithmic differentiation?

• Using the chain rule.

• Using the product rule.

• Replacing f(x) with y.

• Taking the derivative directly.

Which property of logarithms is used to simplify the expression?

• Change of base property

• Power property

• Quotient property

• Product property

What rule is applied to differentiate the natural log of a function?

• Power rule

• Quotient rule

• Product rule

• Chain rule

What is the derivative of natural log u with respect to x?

• 1/u times u'

• u times u'

• 1/u

• u'

How is the expression simplified after applying the product rule?

• By using trigonometric identities

• By factoring out common terms

• By combining like terms

• By expanding the expression

What trigonometric identity is used to simplify the expression?

• Cotangent

• Sine

• Cosine

• Tangent

What is the final step in solving for dy/dx?

• Dividing by y

• Adding y

• Subtracting y

• Multiplying by y

What is factored out in the final expression of the derivative?

• The base of the logarithm

• The smallest common factor

• The greatest common factor

• The exponent of the function

What is the final form of the derivative expression?

• In terms of y only

• In terms of x only

• In terms of both x and y

• In terms of constants only