Derivative of x^x | Summary and Q&A

TL;DR

The derivative of x raised to the x can be found using logarithmic differentiation by replacing y with x raised to the x, taking the natural log, and differentiating both sides.

Key Insights

• ❓ Logarithmic differentiation is a technique used to find derivatives when there are complicated functions involving exponents and logarithms.
• 😑 Using the property of logarithms, the expression can be simplified before differentiating.
• 🙃 Implicit differentiation allows us to find the derivative of both sides of the equation simultaneously.
• 📏 The product rule is used when differentiating the product of two functions.

Transcript

what is the derivative of x raised to the x how can we find the answer to that question well we need to use something called logarithmic differentiation and the first thing we should do is replace or set y equal to x raised to the x that's the first thing you want to do and then take the natural log of both sides so l and y is equal to ln x raised ... Read More

Questions & Answers

Q: How do you find the derivative of x raised to the x using logarithmic differentiation?

To find the derivative of x raised to the x, we use logarithmic differentiation. We start by setting y equal to x raised to the x and take the natural log of both sides. Then, we use implicit differentiation and the product rule to find the derivative.

Q: What is the property of logs used in logarithmic differentiation?

The property of logs used in logarithmic differentiation is that the exponent can be moved to the front. When we take the natural log of y, we can rewrite it as ln y = x times ln x by moving the exponent of x to the front.

Q: How is implicit differentiation used in finding the derivative?

Implicit differentiation is used in finding the derivative by differentiating both sides of the equation with respect to x. We find the derivative of ln y, which is 1/y, and on the right side, we use the product rule to differentiate x times ln x.

Q: What is the final answer for the derivative of x raised to the x?

The final answer for the derivative of x raised to the x is x raised to the x times ln x plus one. This is obtained by simplifying the expression after logarithmic differentiation.

Summary & Key Takeaways

• Logarithmic differentiation is used to find the derivative of x raised to the x.

• By replacing y with x raised to the x and taking the natural log, the expression can be simplified.

• Implicit differentiation and the product rule are then used to differentiate both sides of the equation.