Calculus Derivatives and Logarithms

The video tutorial explains how to find the derivative of a function using logarithmic differentiation. It starts by rewriting the function with rational exponents, then applies logarithmic differentiation by taking the natural log of both sides. The process involves using the chain and product rules to differentiate, followed by solving for dy/dx. The final expression for the derivative is simplified and presented as f'(x).

What is the initial step in rewriting the function f(x) = √x^(7x) using rational exponents?

• Replace f(x) with y

• Express the square root as a power of 1/2

• Multiply the exponents

• Apply the chain rule

After rewriting the function, what is the next step in simplifying it?

• Multiply the exponents

• Divide the exponents

• Subtract the exponents

• Add the exponents

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

• Quotient property

• Change of base property

• Product property

• Power property

Which rule is applied to differentiate the left side of the equation with respect to x?

• Chain rule

• Power rule

• Quotient rule

• Product rule

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

• dy/dx

• 1/x

• y

• 1/y

When applying the product rule, what is the derivative of ln(x) with respect to x?

• ln(x)

• 1/x

• x^2

• x

What is the result of simplifying x/x in the context of this problem?

• x^2

• 1

• 0

• x

What is the final expression for dy/dx in terms of x?

• x^(7x) * (7/2 + ln(x))

• x^(7/2x) * (1 + ln(x))

• x^(7/2) * (7/2 + ln(x))

• x^(7/2x) * (7/2 + 7/2 ln(x))

What is factored out in the final expression for f'(x)?

• 7

• 1/2

• 7/2

• ln(x)

What is the purpose of replacing dy/dx with f'(x) in the final step?

• To factor out common terms

• To simplify the expression

• To match the original question's requirement

• To eliminate y from the equation