# How do you find the first and second derivative of #(lnx)^(x)#?

Let:

The easiest way to differentiate this is to take the natural logarithm of both sides.

Simplify using the logarithm rule:

Differentiate both sides. The left-hand side will need chain rule, similar to implicit differentiation, and the right-hand side will need chain rule and product rule.

The steps to find the second derivative are far too lengthy, and the work is rather pointless, but if you wish try to find it, the second derivative is:

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To find the first derivative of ( (\ln(x))^x ), we use the chain rule:

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

To find the second derivative, we differentiate the first derivative with respect to ( x ) again:

[ \frac{d^2}{dx^2}\left((\ln(x))^x\right) = \frac{d}{dx}\left( x(\ln(x))^{x-1}\left(\frac{1}{x}\right) + (\ln(x))^x\left(\frac{d}{dx}(x\ln(x))\right) \right) ]

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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