Differentiation Techniques for Exponential Functions

The video tutorial covers the process of logarithmic differentiation to find derivatives of functions where a variable is raised to another variable. It includes examples such as differentiating x^x, x^sin(x), ln(x)^x, and x^(1/x). The tutorial explains the use of natural logarithms and the product rule in these processes, providing step-by-step guidance for each example.

What is the first step in using logarithmic differentiation to find the derivative of x raised to the x?

• Set y equal to x raised to the x

• Use the chain rule

• Apply the quotient rule

• Take the derivative directly

When differentiating x raised to the x, what rule is used after taking the natural log of both sides?

• Quotient rule

• Product rule

• Chain rule

• Power rule

In the process of differentiating x raised to the sine x, what is the role of the sine function?

• It is differentiated first

• It is ignored

• It is moved to the front of the natural log

• It is treated as a constant

What is the derivative of sine x when applying the product rule in differentiating x raised to the sine x?

• Natural log of x

• One over x

• Cosine x

• Sine x

What is the first step in differentiating ln x raised to the x?

• Set y equal to ln x raised to the x

• Take the derivative directly

• Use the chain rule

• Apply the quotient rule

In differentiating ln x raised to the x, what is the derivative of ln ln x?

• One over x

• One over x squared

• One over ln x

• One over x ln x

What is the first step in differentiating x raised to the one over x?

• Use the chain rule

• Take the derivative directly

• Set y equal to x raised to the one over x

• Apply the quotient rule

When differentiating x raised to the one over x, what is the derivative of 1 over x?

• Sine x

• Negative one over x squared

• One over x

• Natural log of x

In the final expression for the derivative of x raised to the one over x, what is the simplified form of x raised to the one over x divided by x squared?

• x raised to the one over x minus two

• x raised to the one over x plus two

• x raised to the two over x

• x raised to the negative one over x

What is the final expression for the derivative of x raised to the one over x?

• x raised to the one over x times one minus ln x

• x raised to the two over x times one minus ln x

• x raised to the one over x minus two times one minus ln x

• x raised to the one over x plus two times one minus ln x