How do you find the derivative of # (ln(x))^x#?
Taking Natural Log. of both sides, & using the Rule of Log. Fun, we get,
Now, using the Chain Rule :
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To find the derivative of ( (\ln(x))^x ), we can use the chain rule. Let ( y = (\ln(x))^x ). Taking the natural logarithm of both sides, we get:
[ \ln(y) = x \ln(\ln(x)) ]
Now, differentiate both sides with respect to ( x ):
[ \frac{1}{y} \cdot \frac{dy}{dx} = \ln(\ln(x)) + \frac{x}{\ln(x)} \cdot \frac{1}{x} ]
[ \frac{dy}{dx} = y \left( \ln(\ln(x)) + \frac{1}{\ln(x)} \right) ]
Substitute back ( y = (\ln(x))^x ):
[ \frac{dy}{dx} = (\ln(x))^x \left( \ln(\ln(x)) + \frac{1}{\ln(x)} \right) ]
So, the derivative of ( (\ln(x))^x ) with respect to ( x ) is ( (\ln(x))^x \left( \ln(\ln(x)) + \frac{1}{\ln(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.
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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