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How do you differentiate #f(x)=lnx^x-xlnx #?

Answer 1

William Abdalla

#f(x) = lnx^x-xlnx = 0#, so #f'(x) = 0#

#lnx^x = xlnx#

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Answer 2

Samantha Adkison

To differentiate ( f(x) = \ln(x^x) - x \ln(x) ), we'll use the properties of logarithmic differentiation and the rules of differentiation.

Differentiating term by term:

( \frac{d}{dx} (\ln(x^x)) ): Apply the chain rule and the derivative of natural logarithm. [ \frac{d}{dx} (\ln(x^x)) = \frac{1}{x^x} \cdot \frac{d}{dx}(x^x) ] [ = \frac{1}{x^x} \cdot (x^x \cdot (1 \cdot \ln(x) + x \cdot \frac{1}{x})) ] [ = x^{x - 1} (\ln(x) + 1) ]
( \frac{d}{dx} (x \ln(x)) ): Apply the product rule. [ \frac{d}{dx} (x \ln(x)) = 1 \cdot \ln(x) + x \cdot \frac{1}{x} ] [ = \ln(x) + 1 ]

Now, differentiate ( f(x) ): [ f'(x) = x^{x-1}(\ln(x) + 1) - (\ln(x) + 1) ]

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Answer from HIX Tutor

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.