# Derivative of X^(x^x)?

• appplejack
In summary, the formula for the derivative of x^(x^x) is (x^x)*(1+ln(x)+ln(x^x)). To find the derivative, you can use the chain rule and the product rule. The graph of the derivative is a curve with a vertical asymptote at x=0 and is undefined for negative values of x. The derivative can be simplified to (x^x)*(1+3ln(x)).

## Homework Statement

I uploaded a image file of the question.

## Homework Equations

I've seen the derivative of X^x but how do I do this one?

## The Attempt at a Solution

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Put t=xx and apply the chain rule since you know what d/dx(xsup]x[/sup]) gives.

## 1. What is the formula for the derivative of x^(x^x)?

The formula for the derivative of x^(x^x) is (x^x)*(1+ln(x)+ln(x^x)).

## 2. How do you find the derivative of x^(x^x)?

To find the derivative of x^(x^x), you can use the chain rule and the product rule. First, rewrite the function as e^(x^x*ln(x)). Then, use the chain rule to find the derivative of x^x*ln(x), which is x^x*(1+ln(x)). Finally, use the product rule to find the derivative of e^(x^x*ln(x)), which is (x^x)*(1+ln(x)+ln(x^x)).

## 3. Is the derivative of x^(x^x) defined for all values of x?

No, the derivative of x^(x^x) is not defined for all values of x. It is undefined when x=0, and it does not exist at negative values of x.

## 4. What is the graph of the derivative of x^(x^x)?

The graph of the derivative of x^(x^x) is a curve that starts at (0,1) and approaches 0 as x increases. It has a vertical asymptote at x=0 and is undefined for negative values of x.

## 5. Can the derivative of x^(x^x) be simplified further?

Yes, the derivative of x^(x^x) can be simplified further to (x^x)*(1+ln(x)+2ln(x^x)). This can be rewritten as (x^x)*(1+3ln(x)).