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

Write the function as:

then using the chain rule:

and using the product rule:

Note now that:

so we can simplify as:

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To find the first and second derivatives of (x^{\ln(x)}), we use the chain rule and the product rule.

First Derivative: [ \frac{d}{dx} (x^{\ln(x)}) = \frac{d}{dx} (e^{\ln(x) \cdot \ln(x)}) = e^{\ln(x) \cdot \ln(x)} \cdot (\ln(x) \cdot \frac{1}{x} + \ln(x) \cdot \frac{1}{x}) ]

Second Derivative: [ \frac{d^2}{dx^2} (x^{\ln(x)}) = \frac{d}{dx} \left( e^{\ln(x) \cdot \ln(x)} \cdot (\ln(x) \cdot \frac{1}{x} + \ln(x) \cdot \frac{1}{x}) \right) ]

[ = e^{\ln(x) \cdot \ln(x)} \cdot \left( (\ln(x) \cdot \frac{1}{x} + \ln(x) \cdot \frac{1}{x})^2 + \ln(x) \cdot \frac{-1}{x^2} + \ln(x) \cdot \frac{1}{x} + \ln(x) \cdot \frac{1}{x} \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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