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

The derivative of

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To find the derivative of ( x^{\ln(x)} ), you can use logarithmic differentiation.

Let ( y = x^{\ln(x)} ). Take the natural logarithm of both sides:

( \ln(y) = \ln(x^{\ln(x)}) )

Using the properties of logarithms, we get:

( \ln(y) = \ln(x) \cdot \ln(x) )

Now differentiate both sides with respect to ( x ):

( \frac{1}{y} \cdot \frac{dy}{dx} = \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} )

Substitute ( y = x^{\ln(x)} ) back in:

( \frac{1}{x^{\ln(x)}} \cdot \frac{dy}{dx} = \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} )

Multiply both sides by ( x^{\ln(x)} ):

( \frac{dy}{dx} = x^{\ln(x)} \left( \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} \right) )

Simplify:

( \frac{dy}{dx} = x^{\ln(x)} \left( \frac{\ln(x)}{x} + \frac{\ln(x)}{x} \right) )

( \frac{dy}{dx} = x^{\ln(x)} \cdot \frac{2\ln(x)}{x} )

So, the derivative of ( x^{\ln(x)} ) with respect to ( x ) is ( x^{\ln(x)} \cdot \frac{2\ln(x)}{x} ).

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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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