Derivative of x^x and x^(x^x)

TL;DR

The derivative of x^x^x is x^x^x * ln(x) + x^x^x.

Transcript

we're being asked to find the derivative of y equals x to the x to the x so first we'll find the derivative of x to the x so let y equal x to the x so this is a different y not this one and to find this derivative you just take the natural log of both sides and then you can bring the x downstairs via the power rule so it's ln y equals x times the n... Read More

Key Insights

• 🙃 To find the derivative of x^x, take the natural log of both sides and apply the product rule.
• 🍉 The derivative of x^x^x can be found by first finding the derivative of x^x and then multiplying it with additional terms.
• 📏 The chain rule is applied in finding the derivative of x^x.
• 🐞 Multiplying everything by y is necessary to find dy/dx accurately.
• ❓ The derivative of x^x^x is x^x^x * ln(x) + x^x^x.
• 😑 Simplifying the derivative expression is optional and can be left in the expanded form.
• ☺️ Finding the derivative of x^x^x involves multiple x terms and can be seen as a fun challenge.

Questions & Answers

Q: How do you find the derivative of x^x?

To find the derivative of x^x, take the natural log of both sides, bring the x downstairs using the power rule, and differentiate both sides. The derivative is x^x * ln(x) + x^x.

Q: How is the chain rule used in finding the derivative of x^x?

In finding the derivative of x^x, we use the chain rule by considering y as x^x. Taking the derivative of ln(y) with respect to x gives (1/y) * dy/dx. Then, applying the product rule, we get x^x * ln(x) + x^x as the derivative.

Q: Can you explain the step of multiplying everything by y to find dy/dx?

Multiplying everything by y is necessary to find dy/dx because we considered y as x^x. By multiplying, we end up with dy/dx = y * (x^x * ln(x) + x^x).

Q: Why did you first find the derivative of x^x before finding the derivative of x^x^x?

The reason for finding the derivative of x^x was to build the foundation for finding the derivative of x^x^x. By knowing the derivative of x^x, we could easily apply it in finding the derivative of x^x^x.

Summary & Key Takeaways

• The content explains how to find the derivative of x^x and then uses that information to find the derivative of x^x^x.

• To find the derivative of x^x, take the natural log of both sides and apply the product rule.

• The derivative of x^x^x is x^x^x * ln(x) + x^x^x.