# How do you differentiate #f(x)=x^(ln x)#?

See the explanation section, below.

Differentiate implicitely:

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To differentiate ( f(x) = x^{\ln(x)} ), you can use logarithmic differentiation.

- Take the natural logarithm of both sides: ( \ln(f(x)) = \ln(x^{\ln(x)}) ).
- Apply the logarithmic properties: ( \ln(f(x)) = \ln(x) \cdot \ln(x) ).
- Differentiate implicitly with respect to ( x ): ( \frac{1}{f(x)} \cdot f'(x) = \frac{1}{x} \cdot \ln(x) + \frac{\ln(x)}{x} ).
- Solve for ( f'(x) ): ( f'(x) = f(x) \cdot \left( \frac{\ln(x)}{x} + \frac{1}{x} \right) ).
- Substitute back ( f(x) = x^{\ln(x)} ): ( f'(x) = x^{\ln(x)} \cdot \left( \frac{\ln(x)}{x} + \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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