Calculus Derivatives and Rules

The video tutorial explains how to find the derivative of a variable raised to another variable using logarithmic differentiation. It begins with an introduction to the concept and then provides a detailed step-by-step derivation of x raised to the natural log of x. The tutorial includes example problems, such as finding the derivative of x raised to e to the x and sine x raised to the x, demonstrating the application of logarithmic differentiation and related rules like the chain rule and product rule.

What is the first step in logarithmic differentiation?

• Set the expression equal to y

• Take the derivative of both sides

• Add a constant to both sides

• Multiply both sides by x

Which rule is used to differentiate ln(x^2)?

• Product Rule

• Quotient Rule

• Chain Rule

• Power Rule

How do you simplify the expression 2 ln(x) / x?

• Divide by 2

• Add exponents

• Multiply by x

• Subtract exponents

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

• 1/y

• dy/dx

• 1/y * dy/dx

• y * dy/dx

Which rule is applied when differentiating a product of two functions?

• Product Rule

• Power Rule

• Chain Rule

• Quotient Rule

What is the derivative of e^x * ln(x) using the product rule?

• e^x * ln(x) + e^x/x

• e^x * ln(x) - e^x/x

• e^x * ln(x) * x

• e^x / ln(x)

In the example of x raised to the e^x, what is the first step?

• Add e^x to both sides

• Take the natural log of both sides

• Multiply by e^x

• Subtract e^x from both sides

What is the derivative of sin(x) raised to the x?

• x * ln(sin(x))

• sin(x)^x * (ln(sin(x)) + x * cot(x))

• sin(x) * x

• x * sin(x)

What is the derivative of ln(sin(x))?

• 1/cos(x)

• 1/sin(x)

• sin(x)/cos(x)

• cos(x)/sin(x)

What is the final expression for the derivative of sin(x) raised to the x?

• sin(x)^x * (ln(sin(x)) + x)

• sin(x)^x * (ln(x) + x * cot(x))

• sin(x)^x * (ln(sin(x)) + x * cot(x))

• sin(x)^x * (ln(sin(x)) - x * cot(x))