Derivatives of Logarithmic Functions

The video tutorial explains how to determine the derivative of composite functions using the chain rule. It provides two examples: finding the derivative of f(x) = ln(-4 cosecant x) and g(x) = ln(3 secant x). The process involves identifying the inner function, calculating its derivative, and applying the chain rule to find the derivative of the entire function. Simplification steps are also demonstrated to arrive at the final derivative expressions.

What is the primary rule used to find the derivative of composite functions?

• Power Rule

• Chain Rule

• Quotient Rule

• Product Rule

In the function f(x) = ln(-4 cosecant x), what is the inner function u?

• cosecant x

• -4 cosecant x

• -4

• ln(x)

What is the derivative of cosecant x?

• -secant x tangent x

• secant x tangent x

• -cosecant x cotangent x

• cosecant x cotangent x

After applying the chain rule, what is the expression for f'(x) before simplification?

• 1/u times u prime

• u/u prime

• u prime/u

• 1/u

What is the simplified form of f'(x) for f(x) = ln(-4 cosecant x)?

• -cotangent x

• cotangent x

• -secant x

• secant x

In the function g(x) = ln(3 secant x), what is the inner function u?

• ln(x)

• 3

• 3 secant x

• secant x

What is the derivative of secant x?

• secant x cotangent x

• -secant x cotangent x

• secant x tangent x

• -secant x tangent x

What is the expression for g'(x) before simplification?

• 1/u

• u prime/u

• 1/u times u prime

• u/u prime

What is the simplified form of g'(x) for g(x) = ln(3 secant x)?

• -cotangent x

• tangent x

• -tangent x

• cotangent x

Which of the following is a correct simplification step for g'(x)?

• 3 secant x divided by secant x simplifies to 3

• 3 secant x divided by secant x simplifies to 1

• 3 secant x divided by secant x simplifies to 0

• 3 secant x divided by secant x simplifies to -1