Chain Rule and Logarithmic Derivatives

The video tutorial explains how to find the derivative of a function using the chain rule. It begins by expressing the function with an irrational exponent and identifying the inner function. The derivative of the inner function is calculated, and the chain rule is applied to find the derivative of the composite function. The tutorial then simplifies the expression, converting negative exponents to positive by moving terms to the denominator, and presents the final derivative expression.

What is the first step in finding the derivative of the given function?

• Apply the product rule

• Rewrite the square root using an irrational exponent

• Differentiate directly

• Use the quotient rule

What is the inner function in the composite function?

• x raised to the one half power

• log base three of x

• natural log of x

• x squared

What is the derivative of log base three of x?

• x times natural log three

• natural log of x

• 1 over natural log three times x

• 1 over x

How do you express the original function for applying the chain rule?

• As a sum of two functions

• As a product of two functions

• As u to the power of one half

• As a difference of two functions

What is the expression for the derivative after applying the chain rule?

• One half times u to the power of negative one half times u prime

• u to the power of one half minus one

• u to the power of one half times u prime

• u prime times u squared

What is the next step after finding the derivative using the chain rule?

• Multiply by x

• Integrate the function

• Simplify the expression

• Differentiate again

How do you express log base three of x with a positive exponent?

• As a logarithm

• As a cube root

• As a square root

• As a power of two

What is the final form of the derivative?

• One over 2x natural log 3 times the square root of log base 3 of x

• 2x natural log 3 times log base 3 of x

• One over x squared

• Natural log 3 times x

Why is it important to express the exponent as positive?

• To differentiate again

• To make the function easier to integrate

• To simplify the expression

• To apply the product rule

What mathematical concept is primarily used in this problem?

• Chain rule

• Integration by parts

• Quotient rule

• Product rule