Differentiation and Logarithmic Functions

The video tutorial explains how to find the derivative of a function using logarithmic differentiation. It begins by introducing the function f(x) = 8x^(ln x) and the task of finding its derivative, f'(x), and f'(2). The process involves replacing f(x) with y, taking the natural log of both sides, and applying logarithmic properties to simplify the expression. The tutorial then demonstrates differentiating both sides using the chain rule and solving for dy/dx. Finally, it calculates f'(2) and provides both the exact and decimal approximation of the result.

What is the initial step in rewriting the function for differentiation?

• Replace f(x) with y

• Take the derivative directly

• Use the product rule

• Apply the chain rule

Which logarithmic property is used to separate the terms in the function?

• Product property

• Change of base property

• Quotient property

• Power property

What is the result of applying the power property to the function?

• Natural log of 8

• Natural log of x squared

• Natural log of x times natural log x

• Natural log of x plus y

What rule is applied to differentiate the function with respect to x?

• Product rule

• Quotient rule

• Sum rule

• Chain rule

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

• Natural log of x

• Natural log of y

• 1 divided by y times dy/dx

• y times dy/dx

How is the derivative expressed in terms of x?

• By using the chain rule

• By applying the product rule

• By using the original function

• By replacing y with the original function

What is the value of f prime of x after simplification?

• 16 x to the power of natural log x times natural log x divided by x

• 2 natural log x divided by x

• Natural log of 8

• 8 times x to the power of natural log x

What is the first step in finding f prime of 2?

• Apply the chain rule

• Substitute 2 for x

• Differentiate again

• Use the product rule

What is the exact value of f prime of 2?

• 2 divided by natural log 2

• Natural log of 2

• 16 times 2

• 8 times 2 raised to the power of natural log 2 times natural log 2

What is the decimal approximation of f prime of 2 to four decimal places?

• 7.6543

• 9.1234

• 10.0000

• 8.9655