# What is the derivative of #ln(x)^x#?

We're going to use implicit differentiation to solve this problem.

We'll also be using the product rule and the chain rule .

Know that:

So, say that:

Therefore:

Say that:

Therefore:

This means that:

And as a result...

Now let's differentiate the function you were talking about using implicit differentiation...

And finally...

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The derivative of ( \ln(x)^x ) with respect to ( x ) is given by:

[ \frac{d}{dx} (\ln(x)^x) = x\ln(x)^{x-1}\left(1 + \ln(\ln(x))\right) ]

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