Understanding Derivatives of Composite Functions

The video tutorial explains how to find the derivative of a composite function involving inverse tangent and sine functions. It begins by identifying the outer and inner functions, then applies the chain rule to find the derivative of the inner function. The tutorial continues by simplifying the derivative of the entire function, using the formula for the derivative of the inverse tangent. The process is broken down into clear steps, making it easier to understand the application of the chain rule and the simplification of the resulting expression.

What is the main task given in the problem?

• To determine the derivative of the function

• To find the integral of the function

• To solve the function for x

• To graph the function

What is the outer function in the given composite function?

• Inverse tangent function

• Exponential function

• Cosine function

• Sine function

What substitution is made for the inner function?

• w = 6 inverse tangent

• v = 3x

• u = 8 sine 3x

• t = x squared

What is the derivative of the inner function u with respect to x?

• 3 cosine 3x

• 24 cosine 3x

• 8 sine 3x

• 8 cosine 3x

What is the derivative formula for inverse tangent u with respect to x?

• u prime divided by 1 plus u squared

• 1 divided by u prime

• u squared divided by 1 plus u

• 1 divided by 1 plus u squared

How is the derivative of six times inverse tangent u expressed?

• Six times the integral of u

• Six times u squared

• Six times one divided by the quantity one plus u squared

• Six times the derivative of u

What is the final expression for f prime of x in the numerator?

• 144 sine 3x

• 144 cosine 3x

• 24 sine 3x

• 24 cosine 3x

What is the expression for the denominator in the simplified derivative?

• 1 plus 64 cosine squared 3x

• 1 plus 8 sine squared 3x

• 1 plus 8 cosine squared 3x

• 1 plus 64 sine squared 3x

What mathematical rule is primarily used to find the derivative in this problem?

• Product rule

• Quotient rule

• Chain rule

• Power rule

What is the role of the chain rule in this problem?

• To differentiate the inner function only

• To differentiate both the outer and inner functions

• To differentiate the outer function only

• To find the integral of the function