Differentiation and Logarithmic Functions

The video tutorial explains how to find the derivative of a function using logarithmic differentiation. It starts by introducing the function f(x) = x^(5x) and proceeds to apply logarithmic differentiation. The process involves taking the natural log of both sides, using the power property, and applying the chain and product rules. The derivative is simplified, and the specific value f'(1/2) is calculated and approximated.

What is the initial function given in the problem?

• f(x) = 5x^x

• f(x) = x^(x^5)

• f(x) = x^(5x)

• f(x) = x^5

What is the first step in logarithmic differentiation?

• Apply the chain rule

• Take the natural log of both sides

• Use the quotient rule

• Differentiate directly

Which property of logarithms is used to expand the right side of the equation?

• Change of base property

• Power property

• Quotient property

• Product property

What rule is applied to differentiate the left side of the equation?

• Quotient rule

• Product rule

• Chain rule

• Sum rule

What is the derivative of ln(x) with respect to x?

• x

• x^2

• ln(x)

• 1/x

How is the expression simplified after applying the product rule?

• By combining like terms

• By factoring out x

• By factoring out 5

• By expanding the terms

What is the final expression for f'(x) after simplification?

• x^(5x) * (1 + 5ln(x))

• 5x^(5x) * ln(x)

• x^(5x) * (5 + ln(x))

• 5x^(5x) * (1 + ln(x))

What is the value of f'(1/2) in decimal form?

• 0.2712

• 0.1721

• 0.2172

• 0.1272

What is the exponent in the expression for f'(1/2)?

• 5/2

• 1/2

• 5

• 2

What is the purpose of finding f'(1/2)?

• To check for continuity

• To determine the maximum value

• To find the slope at x = 1/2

• To verify the function