How do you differentiate #f(x)=ln(x)^x#?
Using implicit differentiation , the chain rule , and the product rule ,
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To differentiate the function ( f(x) = \ln(x)^x ), you can use logarithmic differentiation.

Take the natural logarithm of both sides of the equation: ( \ln(f(x)) = \ln(\ln(x)^x) )

Apply the properties of logarithms to simplify the expression: ( \ln(f(x)) = x \ln(\ln(x)) )

Differentiate both sides of the equation with respect to ( x ): ( \frac{1}{f(x)} \cdot f'(x) = 1 \cdot \ln(\ln(x)) + x \cdot \frac{1}{\ln(x)} \cdot \frac{1}{x} )

Solve for ( f'(x) ): ( f'(x) = f(x) \left( \ln(\ln(x)) + \frac{1}{\ln(x)} \right) )

Substitute back the expression for ( f(x) ): ( f'(x) = \ln(x)^x \left( \ln(\ln(x)) + \frac{1}{\ln(x)} \right) )
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When evaluating a onesided 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 onesided 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 onesided 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 onesided 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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