Home > Calculus > Differentiating Logarithmic Functions with Base e

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

Answer 1

We can go by using chain rule and also the rule to derivate exponential functions.

Chain rule states that #(dy)/(dx)=(dy)/(du)(du)/(dx)#. We'll also need the rule to derivate exponential functions, which is: be #f(x)=a^u#, then #f'(x)=a^u(lna)u'#

Let's rename #u=lnx#.

Solving:

#(dy)/(dx)=(lnx)^(x)(ln(lnx))*1#

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Answer 2

Christopher Allers

The derivative of ( (ln(x))^x ) with respect to ( x ) is ( \frac{d}{dx}[(ln(x))^x] = (ln(x))^x \left(\frac{1}{x} + ln(ln(x))\right) ).

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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.